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Shiksha Kendra, 2, Community Centre, Preet Vihar,Delhi-110 092 India
CBSE-i
UNIT-14
CLASS
X
(Core)(Core)
MathematicsSurface Area and VolumeSurface Area and VolumeMathematics
Shiksha Kendra, 2, Community Centre, Preet Vihar,Delhi-110 092 India
CBSE-i
UNIT-14
CLASS
X
(Core)(Core)Surface Area and VolumeMathematics
The CBSE-International is grateful for permission to reproduce
and/or translate copyright material used in this publication. The
acknowledgements have been included wherever appropriate and
sources from where the material may be taken are duly mentioned. In
case any thing has been missed out, the Board will be pleased to rectify
the error at the earliest possible opportunity.
All Rights of these documents are reserved. No part of this publication
may be reproduced, printed or transmitted in any form without the
prior permission of the CBSE-i. This material is meant for the use of
schools who are a part of the CBSE-International only.
The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the learning process in harmony with the existing personal, social and cultural ethos.
The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged. The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in view.
The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand, appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations and additions wherever and whenever necessary.
The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink their approaches for knowledge processing by their learners. In this context, it has become imperative for them to incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of these requirements.
The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all subject areas to cater to the different pace of learners.
The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some non-evaluative components in the curriculum which would be commented upon by the teachers and the school. The objective of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives, SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'. The Core skills are the most significant aspects of a learner's holistic growth and learning curve.
The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework (NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to millions of learners, many of whom are now global citizens.
The Board does not interpret this development as an alternative to other curricula existing at the international level, but as an exercise in providing the much needed Indian leadership for global education at the school level. The International Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The Board while addressing the issues of empowerment with the help of the schools' administering this system strongly recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to their peers through the interactive platforms provided by the Board.
I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr. Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the development and implementation of this material.
The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion forums provided on the portal. Any further suggestions are welcome.
Vineet Joshi
Chairman
PREFACEPREFACE
ACKNOWLEDGEMENTSACKNOWLEDGEMENTS
Advisory Conceptual Framework
Ideators
Shri Vineet Joshi, Chairman, CBSE Shri G. Balasubramanian, Former Director (Acad), CBSE
Sh. N. Nagaraju, Director(Academic), CBSE Ms. Abha Adams, Consultant, Step-by-Step School, Noida
Dr. Sadhana Parashar, Director (Training),CBSE
Ms. Aditi Misra Ms. Anuradha Sen Ms. Jaishree Srivastava Dr. Rajesh Hassija
Ms. Amita Mishra Ms. Archana Sagar Dr. Kamla Menon Ms. Rupa Chakravarty
Ms. Anita Sharma Ms. Geeta Varshney Dr. Meena Dhami Ms. Sarita Manuja
Ms. Anita Makkar Ms. Guneet Ohri Ms. Neelima Sharma Ms. Himani Asija
Dr. Anju Srivastava Dr. Indu Khetrapal Dr. N. K. Sehgal Dr. Uma Chaudhry
Coordinators:
Dr. Sadhana Parashar, Ms. Sugandh Sharma, Dr. Srijata Das, Dr. Rashmi Sethi, Head (I and R) E O (Com) E O (Maths) E O (Science)
Shri R. P. Sharma, Consultant Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO
Ms. Seema Lakra, S O Ms. Preeti Hans, Proof Reader
Material Production Group: Classes I-V
Dr. Indu Khetarpal Ms. Rupa Chakravarty Ms. Anita Makkar Ms. Nandita Mathur
Ms. Vandana Kumar Ms. Anuradha Mathur Ms. Kalpana Mattoo Ms. Seema Chowdhary
Ms. Anju Chauhan Ms. Savinder Kaur Rooprai Ms. Monika Thakur Ms. Ruba Chakarvarty
Ms. Deepti Verma Ms. Seema Choudhary Mr. Bijo Thomas Ms. Mahua Bhattacharya
Ms. Ritu Batra Ms. Kalyani Voleti
English :
Geography:
Ms. Sarita Manuja
Ms. Renu Anand
Ms. Gayatri Khanna
Ms. P. Rajeshwary
Ms. Neha Sharma
Ms. Sarabjit Kaur
Ms. Ruchika Sachdev
Ms. Deepa Kapoor
Ms. Bharti Dave Ms. Bhagirathi
Ms. Archana Sagar
Ms. Manjari Rattan
Mathematics :
Political Science:
Dr. K.P. Chinda
Mr. J.C. Nijhawan
Ms. Rashmi Kathuria
Ms. Reemu Verma
Dr. Ram Avtar
Mr. Mahendra Shankar
Ms. Sharmila Bakshi
Ms. Archana Soni
Ms. Srilekha
Science :
Economics:
Ms. Charu Maini
Ms. S. Anjum
Ms. Meenambika Menon
Ms. Novita Chopra
Ms. Neeta Rastogi
Ms. Pooja Sareen
Ms. Mridula Pant
Mr. Pankaj Bhanwani
Ms. Ambica Gulati
History :
Ms. Jayshree Srivastava
Ms. M. Bose
Ms. A. Venkatachalam
Ms. Smita Bhattacharya
Material Production Groups: Classes IX-X
English :
Ms. Rachna Pandit
Ms. Neha Sharma
Ms. Sonia Jain
Ms. Dipinder Kaur
Ms. Sarita Ahuja
Science :
Dr. Meena Dhami
Mr. Saroj Kumar
Ms. Rashmi Ramsinghaney
Ms. Seema kapoor
Ms. Priyanka Sen
Dr. Kavita Khanna
Ms. Keya Gupta
Mathematics :
Political Science:
Ms. Seema Rawat
Ms. N. Vidya
Ms. Mamta Goyal
Ms. Chhavi Raheja
Ms. Kanu Chopra
Ms. Shilpi Anand
Geography:
History :
Ms. Suparna Sharma
Ms. Leela Grewal
Ms. Leeza Dutta
Ms. Kalpana Pant
Material Production Groups: Classes VI-VIII
1. Syllabus 1
2. Scope document 2
3. Teacher's Support Material 4
Teacher Note 5
Activity Skill Matrix 9
Warm Up W1 10
Dimension of Solids
Warm Up W2 11
Recalling Formulae of Solids
Pre –Content P1 12
Solids Out of Solids
Pre –Content P2 13
Which Deal is Better
Content Worksheet CW1 14
Cubes, Cuboids and Cylinders
Content Worksheet CW2 16
Cones, Spheres and Hemispheres
Content Worksheet CW3 17
Solid Formed by Scooping Out a Part of a Given Solid
Content Worksheet CW4 18
Solids Made up of Combination of Two or More Solids
Content Worksheet CW5 19
Conversion of Solids
Post Content Worksheet PCW1 20
Post Content Worksheet PCW2 20
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Contents
Post Content Worksheet PCW3 20
Post Content Worksheet PCW4 20
Assessment Plan 21
4. Study Material 26
5. Student's Support Material 44
SW1: Warm Up (W1) 45
Dimension of Solids
SW2: Warm Up (W2) 47
Recalling Formulae of solids
SW3: Pre Content (P1) 49
Solids Out of Solids
SW4: Pre Content (P2) 51
Which Deal is Better?
SW5: Content (CW1) 53
Cubes, Cuboids and Cylinders
SW6: Content (CW2) 61
Cones, Sphere and Hemispheres
SW7: Content (CW3) 66
Solid Formed by Scooping out a Part of Given Solid
SW8: Content (CW4) 71
Solids Made up of Combination of Two or More Solids
SW9: Content (CW5) 79
Conversion of Solids
SW10: Post Content (PCW1) 86
SW11: Post Content (PCW2) 88
SW12: Post Content (PCW3) 90
SW13: Post Content (PCW4) 91
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6. Suggested Videos & Extra Readings 92
1
Syllabus –UNIT - 14
Surface Area and Volume (Core)
Recapitulation of known formulae Formula for surface area and volume of cube, cuboid, right circular cylinder, right circular cone, sphere, hemisphere
Combination of Solids Surface area and volume of combination of solids
Conversion of solids Recasting of one solid shape into other solid shape. Application problems
2
Scope Document
Key Concepts
1. Curved Surface Area
2. Lateral Surface Area
3. Total Surface Area
4. Recasting
Learning Objective: Students will be able to
State and use the formula for lateral surface area, total surface area and volume
of cubes and cuboids, when lengths of sides are given.
State and use the formula for curved surface area, total surface area and volume
of a cylinder, when radius and height are given.
Find the surface area (TSA/CSA) of a given cube, cuboid and cylinder when
volume and some other information are given.
Find the volume of a given cube, cuboid and cylinder when surface area and
some other information are given.
State and use the formula for total surface area, curved surface area and volume
of cone, sphere and hemisphere when radius and height are given.
Find the surface area (TSA/CSA) of a given cone, sphere and hemisphere when
volume and some other information are given.
Find the volume of a given cone, sphere and hemisphere when surface area and
some other information are given.
Find the surface area and volume of remaining solid when a portion of it is dug
out or cut out in shape of some other solid.
Find the surface area and volume of the obtained solid when one solid is
converted into another form by recasting.
3
Extension Activities: Students can form nets for prisms and pyramids and can derive
the formula for their lateral surface area and total surface area. Also they can establish
Euler’s Formula.
SEWA: Conduct a survey on packaging of various juices and find which of the
packaging design is best.
Research: According to one study metal available on earth is depleting very fast as the
demands of vehicles are increasing day by day.
Conduct a survey over this and prepare a report to sensitise the people to reduce the
use of individual vehicles.
4
Teacher’s Support Material
5
Teacher’s Note
The teaching of Mathematics should enhance the child’s resources to think and reason,
to visualize and handle abstractions, to formulate and solve problems. As per NCF
2005, the vision for school Mathematics includes.
1. Children learn to enjoy mathematics rather than fear it.
2. Children see mathematics as something to talk about, to communicate through,
to discuss among them, to work together on.
3. Children pose and solve meaningful problems.
4. Children use abstractions to perceive relationships, to see structures, to reason
out things, to argue the truth or falsity of statements.
5. Children understand the basic structure of Mathematics: Arithmetic, algebra,
geometry and trigonometry, the basic content areas of school Mathematics, all
offer a methodology for abstraction, structuration and generalisation.
6. Teachers engage every child in class with the conviction that everyone can learn
mathematics.
Students should be encouraged to solve problems through different methods like
abstraction, quantification, analogy, case analysis, reduction to simpler situations, even
guess-and-verify exercises during different stages of school. This will enrich the
students and help them to understand that a problem can be approached by a variety of
methods for solving it. School mathematics should also play an important role in
developing the useful skill of estimation of quantities and approximating solutions.
Development of visualization and representations skills should be integral to
Mathematics teaching. There is also a need to make connection between Mathematics
and other subjects of study. When children learn to draw a graph, they should be
encouraged to perceive the importance of graph in the teaching of Science, Social
Science and other areas of study. Mathematics should help in developing the reasoning
skills of students. Proof is a process which encourages systematic way of
argumentation. The aim should be to develop arguments, to evaluate arguments, to
6
make conjunctures and understand that there are various methods of reasoning.
Students should be made to understand that mathematical communication is precise,
employs unambiguous use of language and rigour in formulation. Children should be
encouraged to appreciate its significance.
At the secondary stage students begin to perceive the structure of Mathematics as a
discipline. By this stage they should become familiar with the characteristics of
Mathematical communications, various terms and concepts, the use of symbols,
precision of language and systematic arguments in proving the proposition. At this
stage a student should be able to integrate the many concepts and skills that he/she has
learnt in solving problems.
The unit of Surface Area and Volume has high functional aspect and is used from
packaging industry to construction and designing of aeroplanes, from biological
sciences to physical sciences. The activities designed in this unit intend to give exposure
to lots of application problems to the students as well to achieve the following learning
objectives:
• State and use the formula for lateral surface area, total surface area and volume
of cubes and cuboids, when lengths of sides are given.
• State and use the formula for curved surface area, total surface area and volume
of cylinder, when lengths of sides are given.
• Find the surface area (TSA/CSA) of given cubes, cuboids and cylinders when
volume and some other information are given.
• Find the volume of a given cube, cuboid and cylinder when surface area and
some other information are given.
• State and use the formula for total surface area, curved surface are and volume of
cone, sphere and hemisphere when radius and height are given.
• Determine the surface area (TSA/CSA) of given cone, sphere and hemisphere
when volume and some other information are given.
7
• Determine the volume of given cones, spheres and hemispheres when surface
area and some other information are given.
• Determine the surface area and volume of remaining solid when a portion of it is
dug out or cut out in shape of some other solid.
• Determine the surface area and volume of the obtained solid when one solid is
converted into another form by recasting.
• Determine the surface area and volume of frustum of cone under given
conditions.
Although the students are already familiar with all the topics covered under this unit
and have also attained a minimum level of conceptual knowledge as well as procedural
fluency, the purpose of the present unit is to give them more challenging tasks so that
they get opportunity to expand their thinking and reasoning skill.
Warm up activities are designed to help them quickly revise the known formulae as
well to provoke them to think and reflect. This will also enhance the communication
skills of students. They will start thinking loudly and will come out of fear of
mathematics and measurement.
Pre-content activities are extension activities of warm up and learners will observe that
combination of two solids give rise to new solids and will get an idea that the area and
volume of these solids can be calculated. Beauty of this unit also lies in the fact that
students learn to measure without actually measuring. They will develop the skills of
comparing the surface area and volume of two objects.
Some Content worksheets focus on MS excel sheet to let the students to explore the
comparative relation between the two solids when their one or more dimension is
changed.
In all content worksheets, exposure to lots of real life problems are given in order to
prepare them to make effective use of knowledge in life.
8
It is necessary to keep a check on student understanding of conceptual knowledge as
the lack of that often results in applying incorrect formula. Sometimes the learners
are not able to identify when to find lateral surface area and when to find curved
surface area. To overcome the problem teacher should give lots of manipulatives,
demonstrate the statement of the problems using same shaped objects and many
examples from daily life.
Lots of projects can be chosen from real life. For example to create the model of car,
ratio of 1:24 is taken in comparison to actual car. How much metal will be required to
develop the actual car?
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Activity Skills Matrix-Surface Area and Volume
Type of Activity Name of Activity Skill to be developed
Warm UP (W1) Dimension of solids Recognition and recall
Warm UP (W2) Recalling formulae of solids Recognition and memory
Pre-content (P1) Solids out of solids Observation, thinking skills,
analytic skills
Pre-content (P2) Which is a better deal? Analytical skills and
application
Content (CW1) Cubes, cuboids and cylinders Observation, analysis,
inferential problem solving
Content (CW2) Cones, spheres and
hemispheres
Hands on, Observation,
analysis, inferential, problem
solving
Content (CW3) Solid formed by scooping out
a part of a given solid Visualizing, problem solving
Content (CW4)
Solids made up of
combination of two or more
solids
application
Content (CW5) Conversion of solids Observation, Inferential skill,
Thinking Skill, Application
Post-Content (PCW1) Assignment Problem solving skills,
Application
Post-Content (PCW2) MCQ Conceptual understanding,
Problem solving skills.
Post-Content (PCW3) Design and build a model of a
house Application
Post-Content (PCW4) Refresh your memory Memory based
10
Activity 1 – Warm Up (W1)
Dimension of Solids
Specific objective:
To recognise three dimensional objects
Description:
This is a recall exercise. Students will recognize the solids which they have already
learnt in earlier classes. Students will match the picture of the solids with their names.
Execution:
Printed worksheets may be distributed. Students will be given 5 minutes to fill their
worksheets. Cross checking may be done followed by a general discussion.
Parameters for assessment:
Able to recognise given solids.
Able to recall the names of the solids.
Extra reading:
http://www.math-salamanders.com/3-d-shapes.html
http://www.youtube.com/watch?v=ktkzgsA3HZU
http://www.youtube.com/watch?v=TeQD4IRzk2c&feature=related
11
Activity 2 – Warm Up (W2)
Recalling Formulae of Solids
Specific objective:
To recall formulae for surface area and volumes of 3D objects
Description:
Students have already learnt surface area and volumes of solids in class IX. This warm
up is designed for recalling the formulae of surface area and volumes of solids. Students
will play the game and will recall the formulae for playing.
Execution:
Teacher will write the names of solids, their volumes, LSA/CSA, TSA on slips and will
put them in 4 bowls. Teacher will divide the class in four groups. Students of group I
will pick up slips from bowl A, students of group II will pick up slips from bowl B,
students of group III will pick slips from bowl C and students of group IV will pick up
slips from bowl D. Students from group I will try to find the corresponding students
from group II, III and IV. Student who will search his/her partners first will win the
game.
Parameters for assessment:
Able to recognise given solids.
Able to recall formulae for surface area and volumes of various solids.
Extra reading:
http://thinkzone.wlonk.com/Area/AreaVol.htm
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Activity 3 – Pre Content (PC1)
Solids Out of Solids
Specific objective:
To recall and draw dimensions of given solids.
Description:
This is a recall task. Students will draw various dimensions of solids such as length,
breadth and height of cuboid, radius and height of a cylinder etc. This will help
students when they will solve questions on finding surface area and volumes of these
solids.
Execution:
Printed worksheets may be distributed. Students will be given 15 minutes to fill their
worksheets. Cross checking may be done followed by a general discussion.
Parameters for assessment:
Able to recognise various solids.
Able to recognise and draw the dimensions of given solids.
Extra reading:
http://www.attanolearn.com/excel/cbse-10th-math-surface-areas-volumes-volume-combination-solids.jsf
13
Activity 4 – Pre Content (PC2)
Which Deal is Better?
Specific objective:
To introduce combination of solids
Description:
This task is designed to develop an understanding of change in surface area with the
change in shape of a solid, volume remaining the same. Teacher can extend this activity
by conducting hands on activity using sugar cubes. Ask each student to make different
shapes using 27 sugar cubes. Ask them to find the surface area of the shape formed by
them. Note the surface area told by each student on blackboard and conduct a class
discussion.
Execution:
Teacher will pose the problem in front of the students and will give 10 – 15 minutes to
the students to reflect upon their thinking in their notebooks. A general class discussion
should follow thereafter.
For the hands on activity, teacher should give 10 minutes to perform the activity and
then a class discussion should be conducted.
Parameters for assessment:
Able to recall surface area of a cube.
Able to recall area of a square.
Extra reading:
http://www.figurethis.org/challenges/c62/challenge.htm
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Activity 5 – Content Worksheet (CW1)
Cubes, Cuboids and Cylinders
Specific objective :
To state and use the formula for total surface area, lateral surface area and volume of cubes and cuboids, when lengths of sides are given
To state and use the formula for total surface area, curved surface area and volume of cylinder, when radius and height are given.
To find the surface area (TSA/CSA) of a given cube, cuboid and cylinder when volume and some other information are given
To find the volume of a given cube, cuboid and cylinder when surface area and some other information are given
Description:
This is a revision exercise for the students. Task 1 is designed to compare the surface
area volumes of two cylinders when the radius and height of the first is changed to
form the second cylinder. Students will perform hands on activities in groups. Each
group will make a cylinder whose radius is r and height is h with clay. They will note
down the amount of clay used to make the cylinder and the amount of paint used to
paint the cylinder. The group will repeat the same by taking radius as 2r and height as
h/2. The group will again repeat the same by taking radius as 3r and height as h/3.
Based on their activities they will observe the change in surface area and volumes of the
cylinders with the change in their radius and height.
Task 2 is the extension of task 1 and students will answer the questions given in task 2
by performing activities with clay. Task 3 contains questions related to surface area and
volumes of cube, cuboids and cylinders.
Execution:
Task 1 should be done in class in groups. Teacher will divide the class in small groups
and students will do the activity. They will fill the table provided in their notebooks.
15
Task 2 can be given as home assignment followed by a class discussion.
Printed worksheets may be given for task 3.
Parameters for assessment:
Able to make cube, cuboid and cylinder with clay.
Able to find surface area of cube, cuboid and cylinder with given dimensions.
Able to find volume of cube, cuboid and cylinder with given dimensions.
Able to find surface area when volume and some other information is given.
Able to find volume when surface area and some other information is given.
Extra reading:
http://www.youtube.com/watch?v=SmV6i-jVcRU&feature=related
http://www.youtube.com/watch?v=Q0cmYUp3h_w&feature=related
http://www.youtube.com/watch?v=onwM_CSYlu0&NR=1
http://www.youtube.com/watch?v=Incbp5sVxXc&NR=1
http://www.ankn.uaf.edu/publications/VillageMath/cordwood.html
http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/
http://mste.illinois.edu/users/carvell/3dbox/default.html
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Activity 6 – Content Worksheet (CW2)
Cones, Spheres and Hemispheres
Specific objective:
To state and use the formula for total surface area, curved surface area and volume
of cone, sphere and hemisphere when radius and height are given.
To find the surface area (TSA/CSA) of a given cone, sphere and hemisphere when
volume and some other information are given.
To find the volume of a given cone, sphere and hemisphere when surface area and
some other information are given.
Description:
In Task 1, students will take cut outs of plane figures given in the task and will revolve
them to form 3D shapes. Teacher will explain how solids are formed when we revolve a
plain figure along its axis. For example, a right angle triangle when revolved along its
base forms a cone. They will complete the table given in the task.
Task 2 contains questions related to surface area and volumes of cone, sphere and
hemispheres.
Execution:
Task 1 will be done individually by each student and the table should be filled based on
their observation. For task 3, either the teacher can write the questions on black board
and students will solve them in their notebooks or printed worksheets may be
provided.
Parameters for assessment:
Able to find surface area of cone, sphere and hemisphere with given dimensions.
Able to find volume of cone, sphere and hemisphere with given dimensions.
Able to find surface area when volume and some other information are given.
Able to find volume when surface area and some other information are given.
Extra reading:
http://nlvm.usu.edu/en/nav/frames_asid_275_g_3_t_3.html
http://www.youtube.com/watch?v=NAcTBJ1boD4&feature=related
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Activity 7 – Content Worksheet (CW3)
Solids Formed by Scooping out a Part of a Given Solid
Specific objective:
To find the surface area and volume of remaining solid when a portion of it is dug
out or cut out in shape of some other solid.
Description:
Task 1 is designed to arouse critical thinking as to what happens to the surface area and
volume of a solid when a part of it is cut out of it. Students will read the conversation
between Manju and Neelu and will take their stand and will reflect upon the same. Task
2 is based on finding the surface area and volume of remaining solid when a portion of
it is dug out or cut out in shape of some other solid
Execution:
For task 1, teacher should read out the conversation between Manju and Neelu. Teacher
may give 5 minutes for thinking and class discussion follows thereafter. Printed
worksheets may be provided for task 2. Alternatively, students can solve the problems
in their notebooks.
Parameters for assessment:
Able to decide whether the surface area and volume will increase or decrease or
remains same when a portion of solid is cut out.
Able to find the surface area of remaining solid when a portion of it is dug out or cut
out in shape of some other solid.
Able to find the volume of remaining solid when a portion of it is dug out or cut out
in shape of some other solid.
Extra reading:
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Activity 8 – Content Worksheet (CW4)
Solids Made Up of Combination of Two or More Solids
Specific objective:
To find the surface area and volume of solids made up of combination of two or
more solids
Description:
Task 1 is to be played on computers. This task is designed to give an insight of
formation of solids. In task 2, students will find surface area and volumes of solids
formed by combination of cubes. Task 3 is a question framing task in which students
will observe the pictures and will frame questions based on their understanding. Task 4
contains questions based on solids formed by joining two or more solids.
Execution:
Task 1 is to be done in computer lab. Students can work individually or in pairs to do
these tasks. For task 2 and 3, printed worksheets may be provided. Task 4 may be done
in the notebooks followed by a class discussion.
Parameters for assessment:
Able find the surface area of solids made up of combination of two or more solids.
Able find the volume of solids made up of combination of two or more solids.
Extra reading:
http://www.excellup.com/classten/mathten/areavolumeexone.aspx
http://www.excellup.com/classten/mathten/areavolumeextwo.aspx
http://www.youtube.com/watch?v=l5tJ9JocFFM&feature=related
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Activity 9 – Content Worksheet (CW5)
Conversion of Solids
Specific objective:
To find the surface area and volume of the obtained solid when one solid is
converted into another form by recasting.
Description:
Task 1 is designed to make students understand that the volume of a solid remains
same when it is converted from one solid shape to another. Students will make various
solids with the same amount of clay. In task 2 students will measure the dimensions of
the solids made by them and will find their surface area and volumes. Task 3 is to solve
questions based on surface area and volume of the obtained solid when one solid is
converted into another form by recasting.
Execution:
Task 1 is to be conducted in groups. Each group will make solids with clay and will
note down their dimensions. Task 2 may be done in notebooks. A follow up class
discussion should be conducted. For task 3, printed worksheets may be distributed.
Parameters for assessment:
Able to find the surface area of obtained solid when one solid is converted into
another form by recasting
Able to find volume of the obtained solid when one solid is converted into another
form by recasting
Extra reading:
http://www.youtube.com/watch?v=OAotjwOKpQs&feature=related
20
Activity 10- Post Content (PCW1)
Students will be assessed on the worksheet containing questions on finding surface
areas and volumes.
Activity 11- Post Content (PCW2)
Students will be assessed on the worksheet containing MCQ on surface area and
volumes of solids.
Activity 12- Post Content (PCW3)
Students will be assessed on a project work to de done in groups. They will make
design and build model of a house.
Activity 13- Post Content (PCW4)
Students will be assessed on the worksheet containing questions based on formulae
used in the complete chapter.
21
Assessment Plan
Assessment Guidance Plan for Teachers
With each task in student support material a self-assessment rubric is attached for
students. Discuss with the students how each rubric can help them to keep in tune their
own progress. These rubrics are meant to develop the learner as the self motivated
learner.
To assess the students’ progress by teacher two types of rubrics are suggested below,
one is for formative assessment and one is for summative assessment.
Suggestive Rubric for Formative Assessment (exemplary)
Parameter Mastered Developing Needs
motivation
Needs personal
attention
Determine the
surface area
and volume of
given solid.
• Able to find
the surface
area and
volume of
cube and
cuboid when
the length of
edges are
given.
• Able to find
the surface
area and
volume of
cylinder and
cone when
the radius
and height
are given.
• Able to find
the surface
area and
volume of
cube and
cuboid when
the length of
edges are
given .
• Able to find
the surface
area and
volume of
cylinder,
and cone are
given when
the radius
and height
are given.
• Able to state
the surface
area and
volume of
cube and
cuboid when
the length of
edges are
given but
can not find
it accurately.
• Able to state
the surface
area and
volume of
cylinder, and
cone are
given when
the radius
and height
are given but
not able to
find it
accurately.
• Not able to
state the
surface area
and volume
of cube and
cuboid when
the length of
edges are
given.
• Not able to
state the
surface area
and volume
of cylinder,
and cone are
given when
the radius
and height
are given.
22
• Able to find
the surface
area and
volume of
sphere and
hemisphere
when the
radius is
given.
• Able to find
the surface
area when
volume of
solid and any
other
conditions
given.
• Able to find
the volume
when the
surface area
and any
other
condition is
given.
• Able to find
the surface
area and
volume of
sphere and
hemisphere
when the
radius the
radius is
given.
• Not able to
find the
surface area
when
volume of
solid and
any other
conditions
given.
• Not able to
find the
volume
when the
surface area
and any
other
condition is
given.
• Able to state
the surface
area and
volume of
sphere and
hemisphere
when the
radius is
given.
• Not able to
find the
surface area
when
volume of
solid and
any other
conditions
given.
• Not able to
find the
volume
when the
surface area
and any
other
condition is
given.
• Not able to
state the
surface area
and volume
of sphere
and
hemisphere
when the
radius is
given.
• Not able to
find the
surface area
when
volume of
solid and any
other
conditions
given.
• Not able to
find the
volume
when the
surface area
and any
other
condition is
given.
From above rubric it is very clear that
• Learner requiring personal attention is poor in concepts and requires the training
of basic concepts before moving further.
• Learner requiring motivation has basic concepts but face problem in calculations
or in making figures. He can be provided with remedial worksheets.
• Learner who is developing is able to choose suitable method of solving the
problem and is able to get the required answer too. May have the habit of doing
23
things to the stage where the desired targets can be achieved, but avoid going
into finer details or to work further to improve the results. Learner at this stage
may not have any mathematical problem but may have attitudinal problem.
Mathematics teachers can avail the occasion to bring positive attitudinal changes
in students’ personality.
• Learner who has mastered has acquired all types of skills and need more
challenging problems.
24
Teachers’ Rubric for Summative Assessment of the Unit
Parameter 5 4 3 2 1
Surface Area
and Volume
• Able to find the lateral
surface area, total surface
area and volume of cube
and cuboid of given
dimension correctly.
• Able to find the curved
surface area, total surface
area and volume of
cylinder and cone of
given dimension
correctly.
• Able to find the total
surface area and volume
and sphere and
hemisphere of given
radius correctly.
• Able to solve related
problems based on real
life situations.
• Not able to find the
lateral surface area, total
surface area and volume
of cube and cuboid of
given dimension.
• Not able to find the
curved surface area, total
surface area and volume
of cylinder and cone of
given dimension.
• Not able to find the total
surface area and volume
of sphere and
hemisphere of given
radius.
• Not able to solve related
problems based on real
life situations.
Combination
of solids
• Able to find surface area
and volume of solids
obtained by combining
two or more shapes.
• Able to find the surface
area and volume of solid
obtained when one solid
is cut out from another.
• Able to solve related
problems based on real
life situations.
• Not able to find surface
area and volume of
solids obtained by
combining two or more
shapes.
• Not able to find the
surface area and volume
of solid obtained when
one solid is cut out from
another.
• Not able to solve related
problems based on real
life situations.
Conversion
of solids
• Able to understand that
volume of solid remains
same when recasted from
one shape to another, but
• Not able to understand
that volume of solid
remains same when
recasted from one shape
25
surface area may or may
not remain same.
• Able to find the surface
area or unknown
dimension when one
solid is converted into
other.
• Able to solve related
problems based on real
life situations.
to on other, but surface
area may or may not
remain same.
• Not able to find the
surface area or unknown
dimension when one
solid is converted into
other.
• Not able to solve related
problems based on real
life situations.
26
STUDY
MATERIAL
27
SURFACE AREA AND VOLUME (CORE)
INTRODUCTION
You are already familiar with some solid figures such as a cuboid, cube, right circular
cylinder, right circular cone, sphere, hemisphere.
You have also studied finding their surface areas and volumes.
In this unit, we shall first review these concepts and extend this knowledge to study
surface areas and volumes of some more solid figures formed by combination of these
solids.
1 Recapitulation of Surface Areas and Volumes of known solids
Look at the solid (Fig 1(i)).
It is a cuboid.
Its total surface area = 2 (
Lateral surface area = 2 ( + b) h and sq. units and
Volume = b h cu units
Look at the solid (Fig1 (ii))
It is a cube of side (edge) a
Its total surface area = 6a2 sq units
Lateral surface area = 4a2 sq units and
Volume = a3 cu units
Look at the solid (Fig 1 (iii)).
It is a right circular cylinder.
Its curved lateral surface area = 2 rh sq units
Total surface area = 2 r(h + r) sq units
Volume = r2h cu. units
28
Look at the solid (Fig 1 (iv)).
It is a right circular cone.
Its curved (lateral) surface area = r sq units, where
Total surface area = r( +r) sq units, and
Volume = r2h cu units
Look at the solid (Fig 1 (v)).
It is a sphere
Its surface area = 4 r2 sq units,
Volume = r3 cu units
Look at the solid (Fig 1 (vi)).
It is a hemisphere
Its curved surface area = 2 r2 sq units,
Total surface area =3 r2 sq units, and
Volume = r3 cu units Fig. 1
Example 1: Length, breadth and height of a cuboid are 13cm, 5cm and 2cm respectively.
Find its
(i) total surface area (ii) lateral surface area and
(ii) volume.
Solution: Here 13 cm, b = 5 cm and h = 2 cm
(i) Total surface area of the cuboid
= 2( b + bh + h)
= 2 (13 x 5 + 5 x 2 + 13 x 2) cm2
= 2 (65 + 10 + 26) cm2
= 202 cm2
29
(ii) Lateral surface area of the cuboid
= 2 ( +b) h = 2 (13 +5) x 2
= 72 cm2
(iii) Volume of the cuboid = bh
= 13 x 5 x 2 cm2
= 130 cm3
Example 2: Side of a cube is 7 cm. Find its
(i) total surface area , (ii) lateral surface area and
(ii) volume
Solution: (i) Total surface area of the cube = 6a2 = 6 x 7 x 7 cm3 = 294 cm3
(ii) Lateral surface area = 4a2 = 4 x 7 x 7 cm2 = 196 cm2
(iii)Volume = a3 = 7 x 7 x 7 cm3 = 343 cm3
Example 3 : Capacity of cylindrical vessel of height 1 metre, open at the top, is 15.4
litres. Find the area of the metal sheet used in making it. (use )
Solution : Volume of the cylindrical vessel
= 15.4 litres
= 15.4 x 1000 cm3
Let r be the radius of the vessel.
Then Volume = r2h = 15.4 x 1000
x r2 x 100 = 15.4 x 1000 [ 1 m = 100 cm]
r2 = x x
= 49
r = 7cm
Area of the metal sheet used = 2 rh + r2
= 2 x x 7 x 100 + x 7 x 7 cm2
= (4400 + 154) cm2
30
= 4554 cm2
= 0.4554 m2
Example 4: Volume of a right circular cone is 78848 cm3. Its radius is 28 cm. Find its
(i) curved surface area and
(ii) total surface area
Solution: Here base radius (r) = 28 cm.
Volume = 78848 cm3
So r2h = 78848
or x x 28 x 28 x h = 78848
h =
= 96 cm
Slant height =
= cm
= cm
= 100 cm
(i) Curved surface area of the cone
= r
= x 28 x 100 cm2
= 8800 cm2
(ii) Total surface area of the cone = r + r2
= 8800 + x 28 x 28 cm2
= (8800 + 2464) cm2
= 11264 cm2
Example 5: Total surface area of a hemisphere is 462cm2. Find its volume.
Solution: Let the radius of the hemisphere be r.
31
Total surface area = 462cm2
So, 462 = 3 r2 = 3 x r2
r2 =
r2 = 49
r = 7 cm
Thus, Volume of the hemisphere = r3
= x 7 x 7 x 7cm3
= 718.67cm3 (approx)
2. Combination of Solids
So far, we have discussed surface areas and volumes of single known solids only.
But in our daily life, we come across solids which are made up of two or more of
these solids.
For example, see some solids in [Fig2. (i) to (iv)]
32
Fig.2
Each of the above solids is a combination of known solids such as cuboid, cylinder,
cone, hemisphere etc.
Many a times, we need to find surface areas and volumes of such solids.
We explain the process of finding surface areas and volumes of such through some
example.
Example 6: A toy is in the shape of a cone surmounted by a hemisphere as shown in
Fig3. Radius of each of cone and the hemisphere is 5cm and the total height of the toy is
17cm. Find
(i) total surface area, and
(ii) volume of the toy.
Solution:
Total surface area of the toy
= Curved surface area of conical part + Curved surface area of
hemispherical part.
To find curved surface area of conical part
Here r = 5cm, h = height of cone = 17cm – 5cm = 12cm
So, slant height =
33
= cm
= cm
= cm
= 13 cm.
Thus, CSA = r
= . 5. 13 = 65 cm2
To find curved surface area of hemispherical part
r = 5cm.
So, CSA = 2 r2 = 2 (5)2 = 50
From (1), total surface area of the toy = 65 + 50 = 115
Observe that total surface area of the toy is less than the sum of total surface area of the
cone and total surface area of the hemisphere!!
(ii) Volume of the toy
= Volume of conical part + Volume of hemispherical part.
= r2h + r3
= (52) (12) + . 5 x 5 x 5
=
Observe that volume of the toy is the same as the sum of volumes of cone and
hemisphere!
Example 7: A circus tent is cylindrical upto a height of 4.2 m and conical above it. The
common diameter of the base of cylindrical and conical parts is 6m.
If the total height of the tent from the ground is 8.2m, find the cost of canvas needed to
make the tent at the rate of Rs 160 per m2.
Solution: (See Fig 4)
Height of conical part
= Total height of the tent
34
Height of cylindrical part
= (8.2 – 4.2) m = 4m.
Slant height of conical part =
= m
= 5m.
Canvas needed to make the tent
= Curved surface area of cylindrical part + Curved surface
area of conical part
= 2 rh + r
= 2. . 3. (4.2) + . (3) (5) = (79.2 + 47.1) m2 approx
= 126.3 m2
Cost of the tent at the rate of Rs160/m2 = Rs (126.3 x 160) = Rs.
20208
Example 8: A godown is in the shape of a cuboid surmounted
by a half cylinder as shown in Fig 5. Find (i) its capacity and
(ii) the cost of painting it inside except the front rectangular
gate at the rate of Rs. 50/m2 (use = 3.14)
Solution :
(i) Capacity of godown
= Volume of cuboidal part + Volume of half cylindrical part
= bh + r2h
= [(15 x 8 x 10) + (3.14) (4)2 (15)] m3 [Note that diameter = 8m
Height of cylindrical part
= length of cuboidal part]
= (1200 + 376.8)m3
= 1576.8 m3
(ii) Inner surface area of the godown excluding the gate
= Area of 3 walls of the cuboid
m
35
+ (Total curved surface area of the cylinder)
= 2 (15 x 10) m2 + (10 x 8) m2 + [(2 2 r2]
= 380m2 + [(3.14) (4) (15) + (3.14) (4)2] m2
= 380 m2 + (188.40 + 50.24) m2
= 618.64 m2
Thus, cost of painting = Rs. 618.64 x 50 = Rs 30932
Example 9: An oil tanker is in the shape of a cylinder with hemispherical ends as shown
in Fig. 6. If the diameter each of cylinder and hemisphere is 1.4 m, find
(i) capacity of the tanker, and
(ii) the amount of metal sheet required to make the tankers (use ).
Solution
Note that this solid is a combination of two hemispheres and one cylinder. Here
r = 0.7m (Why?)
So, (i) required capacity = r3 + r2h + r3
= r3 + r2h
= m3
= (1.437 + 6.468) m3
= 7.905m3
=7905000 cm3 = 7905 litres
(ii) Metal used = Curved surface area of two hemispheres + Curved surface area of the
cylinder
= 2 r2 + 2 r2 + 2 rh
= 4 r2 + 2 rh
36
= [4 x x 0.7 x 0.7 + 2 x x 0. 7 x 4.2]m2
= (6.16 + 18.48) m2
= 24.64 m2
Example 10 : From a solid cylinder of height 24cm and radius 7cm, a conical cavity of
the same height and same radius is taken out. Find
(i) the volume of remaining solid, and
(ii) total surface area of the remaining solid. (use )
Solution: See Fig 7. Here r = 7cm, h=24cm.
(i) Volume of the remaining solid
= Volume of cylinder - Volume of cone
= r2h – r2h
= r2h = x x 7 x 7 x 24 cm3
= 2464 cm3
(ii) Total surface area of the remaining solid
= Curved surface area of the cylinder
+ Area of base (bottom) of the cylinder
+ Curved surface area of the conical cavity.
= 2 rh + r2 + r
= (7) [2 x 24 + 7 + 25] cm2 [ = = = = 25cm]
= 22 (48 + 7 + 25) cm2
= 1760 cm2
Example 11: A toy rocket is in the shape of a cylinder surmounted by a cone as shown
in Fig.8. The radii of the cylinder and cone are 2cm and 5cm respectively. The total
height of the toy is 42cm. Find the total surface area of the toy. (use )
37
Solution: See Fig 8.
Height of the cylinder = 42 cm – 12cm = 30cm
Radius of cone (r2) = 5cm
Radius of base of the cylinder (r1) = 2cm
Note : is also the radius of the smaller circle in the cone base
Total surface area of the toy = Curved surface area of cylinder +
Area of base of cylinder + Curved surface area of cone
+ Area of base of cone
– Area of base of cylinder
= 2 r1h + 2
1r + + 2
2r - 2
1r
= 2 r1h + r2 + 2
2r , where is the slant height of the cone.
= 2. . 2 (30) + (5) (13) + . 5 x 5
2 2
2 2
= h + r
(12) + (5)
= 169 = 13cm
= 120 + 65 + 25
= 210 = 210 x = 660 cm2
3. Conversion of Solids
Many a times, we need to convert a solid into another solid of the different or of the
same shape. For example, a metallic sphere is converted to form a wire (cylindrical
shape), a cuboidal block to a sphere or a big sphere to small spheres (lead shots etc.).,
the earth dug out from a well to form an embankment (cylindrical shape) etc.
The basic idea behind solving such problem involves equality of volumes of two
objects.
However, in such cases, the surface area of the resulting solid/solids may not be the
same.
Let us explain this through some example.
38
Example 12: The radius of a metallic spherical ball is 3cm. It is melted and recast into
spherical balls each of radius 0.75cm. Find the number of small balls so formed.
Solution: Volume of original metallic ball
= 3 = (3)3 cm3
Volume of a small ball = r3 = (0.75)3 cm3
Let the number of small ball be x
So, volume of small balls = x x ( (0.75)3) cm3
Now, x x (0.75) (0.75) (0.75) = x 3 x 3 x 3
x = = 64
Hence, the number of small balls = 64
Example 13: A cone of height 48cm and base radius 12cm is made up of modelling clay.
It is dismantled and reshaped into a sphere. Find the surface area of the sphere.
Solution: Volume of cone = r2h
= . 12 x 12 x 48cm3
Let R be the radius of the sphere. Then
its volume = R3
Now, R3 = x x 12 x 12 x 48
R3 = (12)3
R = 12
Thus, radius of the sphere = 12cm.
Now, the surface area of the sphere = 4 r2 = 4 x x 12 x 12 = 1810.3 cm3 (approx)
Example 14: A copper rod of radius 0.5cm and length 8cm is drawn into a wire of
uniform thickness cm. Find the length of the wire.
Solution: Volume of rod = r2h = (0.5) (0.5) (8)cm3
Let the length of wire be x cm
39
Note that wire is also a cylinder.
Radius of wire = ( cm) = cm.
Volume of wire = r2h = ( ) ( ) x
So, ( ) ( ) x = (0.5) (0.5) (8)
x = 0.5 x 0.5 x 8 x 30 x 30
= 1800cm
So, the length of wire = 18m
Example 15: Water is flowing at the rate of 2.5km per hour through a cylindrical pipe of
radius 7cm into a rectangular tank of length 25m and 22m wide. Determine the time in
which the level of water in the tank will rise by 35cm.
Solution:
Let the time required for the pipe to raise the water in the tank by 35cm be x hours
Volume of water flowing through the pipe in one hour = r2h
= x x x 2500 m3
[Radius = 7cm = m)
= m3
In x hours water following through the pipe = x m3 (1)
Volume of water in the tank after x hours = Volume of the cuboid of length 25m and
breadth 22m and height 0.35m.
= x b x h
= 25 x 22 x 0.35m3 (2)
From (1) and (2)
x = 25 x 22 x
x = 25 x 22 x x = 5
So required time = 5 hours
40
Example 16: A well of diameter 3.5m is dug 12m deep in a rectangular field of
dimensions 28m x 22m. The earth taken out of it is evenly spread on the remaining part
of the field. Find the height by which the level of field will be raised. (use )
Solution:
Volume of earth dug out
= r2h = (1.75) (1.75) (12)m3
Area of field on which the earth will be spread = (28m x 22m) – (1.75)2 (Why ?)
Let the level of field raised = x m.
So, x [ 28 x 22 – (1.75)2] = (1.75) (1.75) (12)
x = 2
221.75 1.75 12
722
28 1.757
x 22
=
231
277
616 8
= x =
So, the level of the field will be raised by 19 m.
Example 17: A metallic box in the shape of solid cuboid has dimensions
100cm x 50cm x 25cm. It is recast into a solid cube. Find the difference of surface areas
of two solids.
Solution:
Volume of the box = 100 x 50 x 25 cm3
Let the edge of the cube so formed be x cm.
So, x3 = 100 x 50 x 25
= 125000 = (50)3
x = 50
41
Surface area of the box = 2 (100 x 50 + 50 x 25 + 100 x 25) cm2
= 17500 cm2
Surface area of the cube = 6 x (50)2 = 15000 cm2
Difference = (17500 – 15000) cm2 = 2500 cm2
Example 18: A well of inner diameter 7 m has been dug 7 m deep and the earth dug out
is used to form an embankment 4.5 m wide around it. Find the height of the
embankment.
Solution:
Let r be the radius of the well. (See Fig.9)
Then r = 3.5 m
Volume of earth dug out
= r2h = x (3.5) (3.5) (7) m3 [As depth h = 7 m]
= x x x 7 = m3
Outer radius (R) of embankment = 3.5m + 4.5m = 8 m
Area of the base of embankment = Outer circle area – Inner circle area
= (R2 – r2) = (R + r) (R - r)
= (8 + 3.5) (8–3.5) m3
= (11.5) (4.5) m3 = m3
Height of embankment =
=
=
539
22277
14
m
= 1.66 m approx.
42
Example 19: A cylindrical bucket 40 cm high has base radius 36 cm. It is completely
filled with sand. Then it is emptied on the ground forming a conical heap of height
30 cm. Find the radius of the heap and also its curved surface area.
Solution: Radius (r) of the bucket = 36 cm
Height (h) = 40 cm
Volume of sand in bucket = r2h
= (36) (36) (40) cm3 (i)
Let r1 be the radius of conical heap of sand.
Height of conical heap = 30 cm
Volume of conical heap = 2
1r h
= 2
1r x 30 cm3
= 102
1r (ii)
From (i) and (ii),
Volume of conical heap of sand = Volume of sand in bucket
10 = (36) (36) x 40
= (36) (36) (4)
r1 = 36 x 2 = 72
Thus, radius of conical heap of sand = 72 cm
Slant height of the heap =
= cm
= cm = 78 cm
Therefore, curved surface area of the heap = r1
= x 72 x 78 cm2
= 5616 cm2
Example 20: A solid copper sphere of radius 4.9 cm is melted and recast into a solid
cylinder of diameter 14 cm. Find the curved surface area of the cylinder so formed.
Solution: Volume of sphere = r3
43
= (4.9)3 cm3
Let h be the height of the cylinder formed.
Radius of cylinder (r) = = 7 cm.
Since, Volume of sphere = Volume of cylinder formed
So, (4.9)3 = (7)2 h
h = .
h = . = 3.20 cm approx
Curved surface area of the cylinder
= 2 rh
= 2 x x 7 x 3.20 cm2
= 140. 8 cm2 approx
44
Student’s Support Material
45
Student’s Worksheet 1
Warm Up (W1)
Dimension of Solid
Name of Student_______________ Date___________
Make the dimensions of the solids given below and indicate the symbols against each.
Solids Names
h = height
r = radius
46
Self Assessment Rubric 1 – Warm Up (W1)
Parameter
Able to recognise given
solids.
Able to recall the names
of the solids.
47
Student’s Worksheet 2
Warm Up (W2)
Recalling Formulae for Solids
Name of Student_______________ Date_____________
Game: Complete the chain
Bowl A contains slips having names of the solids you learnt in earlier classes. Bowl B
contains slips on which formulae of volumes of these solids are written. Bowl C
contains slips on which formulae of lateral/curved surface area of these solids are
written. Bowl D contains slips on which formulae of total surface area of these solids are
written. Divide your class in four groups. Students of group I will pick up slips from
bowl A, students of group II will pick up slips from bowl B, students of group III will
pick slips from bowl C and students of group IV will pick up slips from bowl D.
Students from group I will try to find the corresponding students from group II, III and
IV. Student who will search his/her partners first will win the game.
Bowl A Bowl B
Bowl C Bowl D
48
Self Assessment Rubric 2 – Warm Up (W2)
Parameter
Able to recognise given
solids.
Able to recall formulae
for surface area and
volume of various solids.
49
Student’s Worksheet 3
Pre Content 1 (PC1)
Solids Out of Solids
Name of Student_______________ Date_____________
Devya made a cube from 27 small cubes. Monika made 2 solids from 27 small cubes as
shown. Both of them used same sized small cubes.
Monika’s Solids Devya’s Solid
They want to paint their solids. Who will need more paint?
Try it yourself and find out.
50
Self Assessment Rubric 3 – Pre Content (PC1)
Parameter
Able to recall surface area
of a cube.
Able to recall area of a
square.
51
Student’s Worksheet 4
Pre Content 2 (PC2)
Which Deal is Better?
Name of Student_______________ Date_____________
Karuna and Rajat went to an ice cream parlour and purchased same flavour ice creams.
Karuna bought ice cream (in a plastic cone of radius 3.5 cm and height 7 cm) for Rs. 20.
Rajat bought ice cream (in a plastic cylinder for height 7 cm and 3.5 cm) for Rs. 20.
Which is a better deal? Why?
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
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52
Self Assessment Rubric 4 – Pre Content (PC2)
Parameter
Able to recall volume of a
cone.
Able to recall volume of a
cylinder.
53
Student’s Worksheet 5
Content Worksheet (CW1)
Cubes, Cuboids and Cylinders Name of Student_______________ Date_____________
Task 1: Activity
Relation between volumes and surface area of two cylinders when the radius of new
cylinder is n times the radius of the original cylinder and height is of the original
cylinder.
Divide the class in small groups of 5 to 6 children each.
Each group will perform the following activity.
With the help of clay, make a cylinder whose radius is r and height is h. Note down the
amount of clay used. Paint the cylinder. Note the amount of paint used.
Now make another cylinder such that its radius is 2r and height is h/2. Again, note the
amount of clay used. Paint the cylinder. Note the amount of paint used.
54
Now make another cylinder such that its radius is 3r and height is h/3. Again, note the
amount of clay used. Paint the cylinder. Note the amount of paint used.
Now, complete the following table.
Gro
up
s
Rad
ius
( r
)
Heig
ht
(h)
Am
ou
nt
of
clay
use
d
Am
ou
nt
of
pain
t
use
d
Rad
ius
(2r
)
Heig
ht
(h
/2)
Am
ou
nt
of
clay
use
d
Am
ou
nt
of
pain
t
use
d
Rad
ius
(3r
)
Heig
ht
(h/3
)
Am
ou
nt
of
clay
use
d
Am
ou
nt
of
pain
t
use
d
Group 1
Group 2
Group 3
Group 4
Group5
Group 6
Observe the above table carefully. Do you see any relation between the amounts of clay
used in the three cylinders made? Is there any relation in the amount of paint used to
paint the three cylinders?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Based on the above observation, comment on the following statements:
55
The volume of a cylinder becomes n times if its radius becomes n times and its height becomes
nth times.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
The surface area of a cylinder remains the same even if its radius becomes n times and its
height becomes nth times.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
Task 2:
Perform the above activity to find out how the volume and surface area changes with
the change in the dimensions of the solids given below:
Cone
Sphere
Hemisphere
You may refer:
Nets (3D model) Generator: http://www.senteacher.org/wk/3dshape.php
56
Task 3 : Do the questions given below:
1. If the length of each edge of a cube is tripled, what will be the change in its volume?
What will be the new surface area?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. A carpenter makes a box which has a volume of 13400 cu. cm. The base has an area
of 670 sq. cm, what is the height of the box? What is the total surface area of the box.
57
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
3. A cuboidal tin open at the top has dimensions of 20 cm X 16 cm X 14 cm. What is the
total area of a sheet of metal required to make 10 such tins?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4. A hollow garden roller 42 cm wide with a girth of 152 cm is made of cast iron 2 cm
thick. Find the volume of iron.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
58
5. The circumference of the base of a right circular cylinder is 176 cm and it is 1 m high.
Find the lateral surface area of the cylinder.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
6. The radius and height of a cylinder are in the ratio 3 : 2 and its volume is 2772 cu. cm.
Find its radius and height. Also, find its curved surface area and total surface area.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
7. The diameter of a roller is 80 cm and its length is 126 cm. It takes 750 revolutions to
level a playground. Find the area of the playground.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
8. The walls and ceiling of a room are to be plastered. The length, breadth and height of
the room are 4.5m, 3m, and 350 cm respectively. Find the cost of plastering at the rate of
Rs 8 per sq. m
______________________________________________________________________________
______________________________________________________________________________
59
______________________________________________________________________________
______________________________________________________________________________
9. 432 cm2 of cardboard is needed to make a cuboidal box which is 8 cm wide and 6 cm
high. Find the breadth of the box. Also, find its volume.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
10. Find the volume of a rectangular solid, each of whose side, front and bottom faces have
an area equal to 12 cm2, 8 cm2 and 6 cm2 respectively.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
11. The length, breadth and height of a cuboid are in the ratio 7 : 4 : 3 respectively. If the
whole surface of the cuboid is 1098 cm2, find its dimensions. Also, find its volume.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
12. 10 cylindrical pillars of a building have to be cleaned. The diameter of each pillar is 50
cm and the height 4 m. Calculate the cost of cleaning all the pillars at the rate of Rs. 5
per sq. m.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
60
Self Assessment Rubric 5 – Content Worksheet (CW1)
Parameter
Able to make cube, cuboid and
cylinder with clay.
Able to find surface area of cube,
cuboid and cylinder with given
dimensions.
Able to find volume of cube,
cuboid and cylinder with given
dimensions.
Able to find surface area when
volume and some other
information is given.
Able to find volume when surface
area and some other information
is given.
61
Student’s Worksheet 6
Content Worksheet (CW2)
Cones, Spheres and Hemishperes
Name of Student_______________ Date_____________
Task 1:
Cut the plane figure given below. Revolve the figures along the vertical axis and the
horizontal axis. Fill the table based on your observations.
Plane figure Draw the
solid
obtained
by
revolving
along
vertical
axis
Name of 3D solid
generated
Measures of
dimensions
Volume
Surface area
Draw the
solid
obtained
by
revolving
along
horizontal
axis
Name of 3D solid
generated
Measures of
dimensions
Volume
Surface area
Solid obtained is
______
r = ……., h = ………
Volume = …………..
Total Surface Area =
…………….
Solid obtained is
______
r = ……., h = ………
Volume = …………..
Total Surface Area =
…………….
Solid obtained is
______
r = ……., h = ………
Volume = …………..
Total Surface Area =
…………….
Solid obtained is
______
r = ……., h = ………
Volume = …………..
Total Surface Area =
…………….
Solid obtained is
______
r = ……., h = ………
Volume = …………..
Total Surface Area =
…………….
Solid obtained is
______
r = ……., h = ………
Volume = …………..
Total Surface Area =
…………….
62
Task 2 : Do the following
1. How many meters of cloth 5m wide will be required to make a conical tent, the
radius of whose base is 7m and whose height is 24m?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. The radius of a sphere is 7cm. If the radius be increased by 50%, find by how much
percent its volume is increased.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
3. The internal and external diameters of a hollow hemispherical vessel are 24cm and
25cm respectively. If the cost of painting 1cm2 of the surface area is Rs. 5.25, find the
total cost of painting the vessel all over
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4. The volume of a vessel in the form of a right circular cylinder is and its
height is 7cm. Find the radius of its base. Also, find its curved surface area and total
surface area.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
63
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
5. A closed iron tank with the dimensions 12 m x 9 m x 4 m is manufactured.
Determine the cost of iron sheets used at the rate of Rs. 5 per metre. (The sheet is 2 m
wide).
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
6. How many metres of cloth 1.1m wide will be required to make a conical tent whose
vertical height is 12cm and base radius is 16m? Find also the cost of the cloth used at
the rate of Rs. 14 per metre.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
_________________________________________________________________________
7. A right circular cone of height 4 cm has a curved surface area 47.1 cm2. Find its
volume.
___________________________________________________________________________
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___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
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___________________________________________________________________________
8. If the surface area of a sphere is 616cm2, find its volume.
___________________________________________________________________________
___________________________________________________________________________
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64
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
9. The circumference of the edge of a hemispherical bowl is 132cm. Find the capacity of
the bowl.
___________________________________________________________________________
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___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
10. A cone and a hemisphere have equal bases and equal volumes. Find the ratio of
their heights.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
65
Self Assessment Rubric 6 – Content Worksheet (CW2)
Parameter
Able to find surface area
of cone, sphere and
hemisphere with given
dimensions.
Able to find volume of
cone, sphere and
hemisphere with given
dimensions.
Able to find surface area
when volume and some
other information are
given.
Able to find volume
when surface area and
some other information
are given.
66
Student’s Worksheet 7
Content Worksheet (CW3)
Solids Formed after Scooping Out a Part of a Given Solid
Name of Student __________________ Date_____________
Task 1: Manju and Neelu are doing their math homework on surface area and volumes.
Read their conversation carefully.
Manju: Whenever we scoop out a solid from another solid, the surface area and volume
of the new solid thus formed is always less than the original solid.
Neelu: I don’t agree Manju. I think only the volume will be less.
Manju: But we are taking out some amount of solid, why will the surface area not
decrease?
Reflect your views on the above discussion.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Task 2: Do the following
1. A plot of land measures 240 m x 180 m. A trench, 10 m wide, is dug around it. The
earth dug out is spread evenly over the plot. This raises the level of the plot by 25
cm. Find the depth of the trench.
67
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
2. Find the area to be painted in the following box with a cylindrical hole. Given
that length = 15 cm, width = 12 cm, height = 20 cm and radius = 3 cm.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
68
3. How many cubic metres of earth must be dug to sink a well 7 metres deep and
diameter 28 m?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4. The dimensions of a field are 160 m x 130 m. A pit is dug in the field, which is 40
m long, 20 m wide and 4 m deep. The earth removed is spread over the remaining
area of the field. Calculate the level of the field raised.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
5. A well is dug 20 m deep. Its diameter is 14 m. The earth dug out is spread evenly
to form a platform 22 m x 14 m. Determine the height of the platform.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
69
6. The largest sphere is carved out of a cube of side 7cm. Find the volume of the
sphere.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
7. A cylinder is 14 cm in length. The difference between the inside and the outside
surfaces of this cylinder is 88 cm2. If the volume of the cylinder is 176 cm3,
calculate its inner and outer radii.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
70
Self Assessment Rubric 7 – Content Worksheet (CW3)
Parameter
Able to decide whether
the surface area and
volume will increase or
decrease or remains same
when a portion of solid is
cut out.
Able to find the surface
area of remaining solid
when a portion of it is
dug out or cut out in
shape of some other solid.
Able to find the volume
of remaining solid when a
portion of it is dug out or
cut out in shape of some
other solid.
71
Student’s Worksheet 8
Content Worksheet (CW4)
Solids Made up of Combination of Two or More Solids
Name of Student __________________ Date_____________
Task 1: Rotating Houses
Go to the link: http://www.mathsnet.net/geometry/solid/rotatinghouses.html
72
Task 2:
Find the surface area & volume of the following figures.
Solids Volume Surface Area
73
Task 3:
Frame questions involving the application of surface area and volumes based on the
pictures given below:
Pictures Write question based on the picture
74
Task 4: Do the following
1. Part of an iron pillar is in the form of a right circular cylinder and remaining is in the
form of a right circular cone. The radius of the base of the cone and the cylinder is 8
cm. The cylindrical part is 240 cm high and the conical part is 36 cm high. Find the
weight of the pillar, if 1 cm3 of iron weighs 7.8 gm.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
2. A company manufactures toys of the following
shape. The height of left conical part is 7 cm and
radius is 3.5 cm. The radius of the cylindrical part is 7
cm and the height is 2 cm. The total height of the toy
75
is 12.5 cm. How much paint is required to paint 225 such toys?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
3. Three identical double napped cones are placed one above the
other to form a decorative piece, which stands on a cylindrical
base. The total height is 25 inch and the height of cylinder is 4
inch. The radius of the cones is 3.5 inch and that of the cylinder is
6 inch. What is the total volume of the piece?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
_____________________________________________________________________________
4. Writing Maths
Explain, how will you find the amount of polish required to polish the vessels below all
over.
76
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
77
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
78
Self Assessment Rubric 8 – Content Worksheet (CW4)
Parameter
Able to find the surface
area of solids made up of
combination of two or
more solids.
Able to find the volume
of solids made up of
combination of two or
more solids.
79
Student’s Worksheet 9
Content Worksheet (CW5)
Converssion of Solids
Name of Student _________________ Date_____________
Task 1: Take 200 gm of clay. Make a cylinder from this clay (use all clay). Note down
the dimension of your cylinder.
Now, make a cone from the cylinder using all the clay. Note the dimension of the cone.
Again change the cone to a sphere using all the clay. Note the radius of the sphere.
Change the sphere into a cube and then to a cuboid using the clay completely. Note
their dimensions also.
Rajan claimed that the surface area of all these will be same. Reflect.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Ritu claimed that their volume will be same. Reflect.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Task 2: Based on the solids made above, complete the table given below:
Solids made Dimensions Volume Surface area
Cylinder
80
Cone
Sphere
Cube
Cuboid
81
When one solid is converted into another, its volume is conserved but surface area may change.
Reflect.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
_____________________________________________________________________________
Task 3: Do the following questions:
1. Three metal cubes with edges 3 cm, 4 cm and 5 cm are melted to form a single cube.
Find the edge of this new cube.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
82
2. A measuring jar of internal diameter 10 cm is partially filled with water. Four equal
spherical balls, each of diameter 2 cm, are dropped in it. The balls sink to the bottom
of the jar. What will be the change in the level of the water in the jar?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
3. A river 2 m deep and 45 m wide is flowing at the rate of 3 km per hour. Find the
amount of water that runs into the sea per minute.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
83
4. How many cubes of side ½ cm are required to build a cube of volume 8 cu.cm.?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
5. A concrete truck arrives at a job site holding 7.8 m3 of concrete. If the platform being
constructed is 5 m across and 2 m thick, how long will the platform be if constructed
from the amount of concrete on the truck?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
6. A conical vessel whose internal radius is 5cm and height 24cm is full of water. The
water is emptied into a cylindrical vessel with internal radius 10cm. Find the height
to which the water rises in the cylindrical vessel.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
84
___________________________________________________________________________
__________________________________________________________________________
7. The diameter of a sphere is 42 cm. It is melted and drawn into a cylindrical wire of
28 mm diameter. Find the length of the wire.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
8. A solid metallic cylinder of radius 14cm and height 21cm is melted and recast into 72
equal small spheres. Find the radius of one such sphere.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
85
Self Assessment Rubric 9 – Content Worksheet (CW5)
Parameter
Able to find the surface
area of solid obtained
when one solid is
converted into another
form by recasting
Able to find volume of
the solid obtained when
one solid is converted
into another form by
recasting
86
Student’s Worksheet 10
Post Content (PCW1)
Name of Student _________________ Date_____________
1. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The
diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm.
Find the inner surface area of the vessel.
2. A conical vessel with internal radius 6 cm and height 8 cm is completely filled with
water. A sphere is lowered into the water and its size is such that when it touches
the sides, it is just immersed. Find the volume of water overflows.
3. A given amount of wax in cylindrical form is heated in a metal container and then
poured into another container. A new candle is formed which is shaped like a fish.
Fill in the blanks
(a) The volume of the wax _______________ (is changed/remains same)
(b) The total surface Area ________________ (is changed/remains same)
4. How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be
melted to form a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?
5. A wooden ariticle was made by scooping out a hemisphere from each end of a
solid cylinder. If the height of the cylinder is 10 cm and its base is of radius 3.5 cm,
find the total surface area of the article.
6. A sphere of radius 3 cm is dropped into a cylindrical vessel partly filled with
water. The radius of the vessel is 6 cm. If the sphere is submerged completely, by
how much will the surface of water be raised?
7. The radii of the ends of the frustum of a cone are 14 cm and 21 cm and the slant
height is 8 cm. Find the area of the curved surface.
87
8. Find the volume of a frustum of a cone whose face radii are 7 m and 4m and height
is 4m.
9. A lamp shade made of a special chart paper is in the form of a frustum of a cone
open at both ends. The radii of its ends are 16 cm and 24 cm and its height is 6 cm.
Find the cost of paper used if one square m costs Rs 0.70.
10. A solid sphere of radius 6 cm is melted into a hollow cylinder of uniform
thickness. If the external radius of the base of the cylinder is 5 cm and its height is
32 cm, find the uniform thickness of the cylinder.
88
Student’s Worksheet 11
Post Content (PCW2)
Name of Student _________________ Date_____________
Multiple Choice Questions
1. The total surface area of a solid hemisphere of radius r is
(A) r2 (B) 2 r2 (C) 3 r2 (D) 4 r2
2. The volume and the surface area of a sphere are numerically equal, then the radius
of sphere is
(A) 0 units (B) 1 units (C) 2 units (D) 3 units
3. A cylinder, a cone and a hemisphere are of the same base and of the same height.
The ratio of their volumes is
(A) 1 : 2 : 3 (B) 2 : 1 : 3 (C) 3 : 1 : 2 (D) 3 : 2 : 1
4. Small spheres, each of radius 2cm, are made by melting a solid iron ball of radius 6
cm, then the total number of small spheres is
(A) 9 (B) 6 (C) 27 (D) 81
5. A solid sphere of radius r cm is melted and recast into the shape of a solid cone of
height r. Then the radius of the base of cone is
(A) 2r (B) r (C) 4r (D) 3r
6. Three solid spheres of diameters 6 cm, 8 cm and 10 cm are melted to form a single
solid sphere. The diameter of the new sphere is
(A) 6 cm (B) 4.5 cm (C) 3 cm (D) 12 cm
7. The radii of the ends of a frustum of a cone 40 cm high are 38 cm and 8 cm. The
slant height of the frustum of cone is
(A) 50 cm (B) 10√7 cm (C) 60.96 cm (D) 4√2 cm.
8. A 4 cm cube is cut into 1 cm cubes. What is the percentage increase in the surface
area after such cutting?
(A) 4% (B) 300% (C) 75% (D) 400%
89
9. If we reduce the height of a cylinder by ¼ and double the radius, which of the
following will hold true?
A. Volume of the cylinder will double
B. Volume of the cylinder will remain unchanged
C. Volume of the cylinder will reduce to ½
D. None of the above
10. If we reduce the height of a cylinder by ¼ and double the radius, which of the
following will hold true?
A. Lateral Surface Area of the cylinder will double
B. Lateral Surface Area of the cylinder will remain unchanged
C. Lateral Surface Area of the cylinder will reduce to ½
D. None of the above
90
Student’s Worksheet 12
Post Content (PCW3)
Name of Student _________________ Date_____________
Design and build a model of a house
Work in groups to make a model of a house.
Hints and Suggestions:
The scale should be 1 inch = 4 feet
Material to be used: Paper, wood, etc.
The model should not exceed beyond ½ sq. mt.
All three dimensional solid objects should be used. Namely Cube, Cuboid, Cylinder, Sphere, hemi-sphere, cone and frustum.
Students are expected to find innovative ways to use the objects in innovative/creative ways.
(Fig. 1)
Expectations from the project report:
1. Architectural layout of the model house
2. A picture of the model prepared
3. Cost of painting the model house externally
4. Volume of material used in building the model house
91
Student’s Worksheet 13 Post Content (PCW4)
Name of Student _________________ Date_____________
Refresh Your Memory
Solid Draw
Figure
Write
Dimensions
Formula of
Volume
Formula of
LSA/CSA
Formula of
TSA
Cube
Cuboid
Cylinder
Hollow
Cylinder
Cone
Sphere
Hemisphere
Hemispherical
bowl of certain
thickness
92
Suggested Videos & Extra Readings
http://www.youtube.com/watch?v=SmV6i-jVcRU&feature=related
http://www.youtube.com/watch?v=Q0cmYUp3h_w&feature=related
http://www.youtube.com/watch?v=onwM_CSYlu0&NR=1
http://www.youtube.com/watch?v=Incbp5sVxXc&NR=1
http://www.ankn.uaf.edu/publications/VillageMath/cordwood.html
http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/
http://mste.illinois.edu/users/carvell/3dbox/default.html
http://nlvm.usu.edu/en/nav/frames_asid_275_g_3_t_3.html
http://www.youtube.com/watch?v=NAcTBJ1boD4&feature=related
http://www.excellup.com/classten/mathten/areavolumeexone.aspx
http://www.excellup.com/classten/mathten/areavolumeextwo.aspx
http://www.youtube.com/watch?v=l5tJ9JocFFM&feature=related
http://www.youtube.com/watch?v=OAotjwOKpQs&feature=related
http://www.youtube.com/watch?v=glr5OhksSmw&feature=relmfu
http://www.youtube.com/watch?v=SdgV94_HfxI&feature=related
http://www.youtube.com/watch?v=aefdZW3htIw&feature=related
http://www.youtube.com/watch?v=yioM-LCyh0k&NR=1
http://thinkzone.wlonk.com/Area/AreaVol.htm
http://www.figurethis.org/challenges/c03/challenge.htm
http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/
http://www.beaconlearningcenter.com/weblessons/solidpatterns/default.htm#page1
http://www.figurethis.org/challenges/c62/challenge.htm
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CENTRAL BOARD OF SECONDARY EDUCATION