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Aim: Volume Course: Applied Geo.
Do Now: What is the area, in terms of of the shaded region in the figure below?
Aim: The Third Dimension: Volume – What is it?
sA. of sq. = s2
A. of circle = r2
A. of shaded region = A. of sq. - A. of circle
A. of s. r. = s2 - 1/2s)2 = s2 - (1/4 s2)= s2 - 1/4s2
r = 1/2 s
Aim: Volume Course: Applied Geo.
Perimeter & Area
PerimeterPerimeter - the distance around a polygon PP = 1 + 2 + 3 + 4
AreaArea - the space inside a polygon - measured in square units
1
1 sq. un.
1
1 sq. un.
1
1 sq. un.
1
1 sq. un.
1
1 sq. un.
1
1 sq. un.
11
12
3
4
units2, in2, miles2, etc.
Aim: Volume Course: Applied Geo.
Volume
VolumeVolume - the measure of the space inside a polyhedron -measured in cubic units.
V = Bh, where B = the area of the base
11 1
1 cu. unit
4
12
8 cubic units.
27cubic units.
3
3
3
Polyhedron - three dimensional figure whose surfaces are polygons
Edge – a segment that is the intersection of two faces
Vertex – point where edges meet
Aim: Volume Course: Applied Geo.
Volume Formulas
Prism – a polyhedron with two congruent, parallel bases. The other faces are lateral faces.
hBV Prism
wl
h
Rectangular B = l • w
Triangular B = 1/2 bh
B - area
of the baseb
Rectagular prism – V = l w h
A triangular prism is a solid whose base is a triangle.
Aim: Volume Course: Applied Geo.
Model Problems
Find the Volume of the following polyhedron: 13
17
10
V = l • w • h = 17 x 13 x 10 = 2210 units3
hwlV
Prism
wl
h
Rectangular B = l • w
Aim: Volume Course: Applied Geo.
Model Problem
What is the volume of a rectangular box whose base measures 3 units by 10 units and whose height is 20 units?
Prism
wl
h
Rectangular B = l • w
Rectangular prism – V = l w h
20
310
V = 10 3 20 = 600 units3
Aim: Volume Course: Applied Geo.
Model Problem
The volume of a rectangular box is 2,400 cu. units. The measurements of the base are 60 x 20. What is the height of the box?
wl
hBV = 2400 units2
2400 = 60 x 20 x h
2400 = 1200 x h
2 = h
60
h
20
Aim: Volume Course: Applied Geo.
Model Problem
The base of a triangular prism has an area of 15 cm2 and a height of 30 cm. What is the volume of the triangular prism?
Triangular prism – V = B h V = 15cm2 30 = 450 units3
hBV
Triangular B = 1/2 bh
B - area
of the baseb
30
15 cm2
Aim: Volume Course: Applied Geo.
Volume Formula
Prism – a polyhedron with two congruent, parallel bases. The other faces are lateral faces.
3eV
Cube
ee
e
What is the volume of a cube with a side of 10 units?
10
1010
3 310 1000V unit
Aim: Volume Course: Applied Geo.
Model Problems
A cylindrical tank has a radius of 10 ft. and a height of 25 ft. What is its volume? Give your answer in terms of
hrV 2
Cylinderr
h
r
h 10 ft25 ft
210 25 2500V
Aim: Volume Course: Applied Geo.
Volume Formulas
h
B = area of base of pyramid
hBV 31
Pyramid
Pyramid – 3 dimensional figure with a single base and sides that are triangles.
Triangular pyramid
Rectangular pyramid
Pentagonal pyramid
Aim: Volume Course: Applied Geo.
Model Problems
Find the volume of the pyramid with a height of 9.
15
10
9
hBV 31
)91015(31 V
B = l • w )1350(31V
3450 unitsV
Aim: Volume Course: Applied Geo.
Volume Formula
hrV 2
31
Cone
Cone – a pyramid with a circle for a base
What is the volume of a cone whose height is 10 units and whose base has radius 5? Express in terms of
h = 10
r = 5
21 5 103V
Aim: Volume Course: Applied Geo.
Model Problems
The main tank at the Living Seas Aquarium at EPCOT Center in Florida is the largest enclosed tank in the world. It is a cylinder with diameter 203 ft. and height 25 ft. About how many million gallons of water does this tank hold? (1 gal. = 231 in3; 1728 in3 = 1 ft3)
25 ft.
203 ft.
Radius is 1/2 the diameter r = 203 2 = 101.5 ft.
hrV 2
h = 25’
)25()5.101(14.3 2V3625.726,808 ft
33
33 608,479,397,11728625.726,808 inft
inft
gallonsgalinin 273.695,049,6231608,479,397,1
33
Aim: Volume Course: Applied Geo.
A. Find the surface area of the Great Pyramid, including its base.
Model Problems
The Great Pyramid at Giza, Egypt, was built about 2580 B.C. as a final resting place for Pharoah Khufu. At the time it was built, its height was about 481 ft. Each edge of the square base was 756 feet long.
Height 481 ft.
756 ft 756’
B. Find the volume of the Great Pyramid.
Aim: Volume Course: Applied Geo.
Model Problems
Zia is planning to landscape her backyard. The yard is a 70ft-by-60ft rectangle. She plans to put down a 4- in. layer of topsoil. She can buy bags of topsoil at $2.50 per 3-ft3 bag, with free delivery. Or she can buy bulk topsoil for $25.00 per yd3, plus a $20 delivery fee. Which option is less expensive. Show your calculations and explanation. (1 yd3 = 9 ft3)
70 feet
60’4”
Aim: Volume Course: Applied Geo.
Model Problems
A cylinder has be cut out of the figure below. Find the volume of the remaining figure. Round your answer to the nearest tenth.
4”
6 in. 6 in.
6 in.