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Unit 4: Surface Area and Volume of Solids
Lesson 4.1 Explore Solids Lesson 12.1 from textbook
Objectives: • Determine which of the different three-dimensional figures or solids are polyhedra
• Identify parts of the polyhedron such as faces, vertices, and edges.
• Identify the solids formed by taking cross sections of a polyhedron.
TYPES OF SOLIDS
CLASSIFYING SOLIDS
To classify a polyhedron, ______________________________________________________.
edge
vertex
face
Face __________________________________________
Edge __________________________________________
Vertex _________________________________________
Prism
Pyramid
Polyhedra
Cone
Cylinder
Not Polyhedra
Regular, Convex Polyhedron
Nonregular, Concave Polyhedron
Polyhedron
Type of Polyhedron
Shape of Base
Name of Polyhedron
Example 1
Tell whether the solid is a polyhedron. If it is, name the polyhedron and find its number of faces,
vertices, and edges.
______________________________________________________
Euler’s Theorem
The number of faces, vertices, and edges of a polyhedron are related by the formula
______________________.
CROSS SECTIONS
Cross Section ________________________________
Example 2
Describe the shape formed by the intersection of the plane and the cube.
___________________ ____________________
Unit 4.2: Surface Area and Volume of Solids
Lesson 4.2 Surface Area of Prisms and Cylinders Lesson 12.2 from textbook
Objectives: • Derive the formula for surface area of a prism and a cylinder using the area formulas for
parallelograms and circles.
• Use the formulas to calculate the surface area of a prism and a cylinder.
• Use the surface area of a prism or cylinder to find the unknown dimensions.
PRISMS
Name of Prism ______________________________________
Number of Bases ____________________________________
Number of Lateral Faces ______________________________
NET OF A RECTANGULAR PRISM
Lateral Area = _____________ Base Area = _______________ Surface Area = _________________
OBLIQUE AND
RIGHT PRISMS
Surface Area of a Right Prism Theorem Example 1
P = perimeter of base, B = base Area, h = height of prism Find the surface area of the right prism.
S = __________________________
S = ___________________________
CYLINDERS NET OF A CYLINDER
Shape of Base _______________ Base Area = _______________ Lateral Area = _______________
Surface Area = ______________________
Surface Area of a Cylinder Theorem Example 2
r = radius of base, h = height of the cylinder Find the surface area of the cylinder.
S = ____________________________ S = ____________________________
Example 3
Find the height of the right cylinder which has a
surface area of 157.08 square meters.
h = ___________________________
Unit 4: Surface Area and Volume of Solids
Lesson 4.3 Surface Area of Pyramids and Cones Lesson 12.3 from textbook
Objectives: • Derive the formula for surface area of a pyramid and a cone using the area formulas for
parallelograms, triangles and circles.
• Use the formulas to calculate the surface area of a pyramid and a cone.
• Use the surface area of a pyramid or cone to find the unknown dimensions.
PYRAMIDS
NET OF A PYRAMID
l = _____________________
b = ____________________
Lateral Area _____________________________
Surface Area = ___________________________
Surface Area of a Pyramid Theorem Example 1
B = base area, P = perimeter of the base, l = slant height Find the lateral area of the pyramid.
S = _________________________ L = ____________________________
Example 2 Find the surface area of the pyramid.
S = ____________________________
CONES NET OF A RIGHT CONE
Base Area ______________________
Lateral Area ____________________
Surface Area ____________________
Surface Area for Cones Theorem
r = radius of the base, l = slant height of cone
S = _________________
Example 3
Find the surface area of the cone.
S = ___________________________ S = ___________________________
Unit 4: Surface Area and Volume of Solids
Lesson 4.4: Volume of Prisms and Cylinders Lesson 12.4 from textbook
Objectives: • Use the formulas to calculate the volume of a prism and a cylinder.
• Use the volume of a prism or cylinder to find the unknown dimensions.
• Solve problems involving unit conversion for situations involving volumes within the same
measurement system.
Volume of a Cube Postulate Volume Congruence Postulate
The volume of a cube is the cube of If two polyhedra are congruent, then
the length of its side.
their volumes _____________________________.
Volume Addition Postulate.
V = _______________________ The volume of a solid is the sum of the volume of
its non-overlapping parts.
Volume of a Prism Volume of a Cylinder
B = Base area, h = height of prism B = Base area, h = height of prism
V = _______________________ V = Bh = _________________________
*The shape of the base could be a square,
rectangle, trapezoid, triangle, or any polygon.
Example 1
Find the volume of the solid.
V = ______________________ V = _____________________
Example 2
If the volume of the cube is 90 in3, find the length of each side x.
x = ________________
Cavalieri’s Principle Example 3
Find the volume of the solid.
All three solids have equal heights and cross-sectional areas B.
Bonaventura Cavalieri stated that:
____________________________________ ________________________
Example 4 Example 5
Find the volume of the solid. The sculpture is made of 13 beams. The
dimensions of each beam are 30 by 30 by 90. Find
its volume.
V = ____________________
V = ____________________
x x
x
Unit 4: Surface Area and Volume of Solids
Lesson 4.5 Volume of Pyramids and Cones Lesson 4.5 from textbook
Objectives: • Use the formulas to calculate the volume of a pyramid and cone.
• Use the volume of a pyramid and cone to find the unknown dimensions.
• Solve problems involving unit conversion for situations involving volumes within the same
measurement system.
Volume of a Pyramid Theorem Volume of a Cone Theorem
V = _____________________________ V = _____________________________
Example 1
Find the volume of the solid.
V = _________________ V ≈ __________________
Example 2
If a square base pyramid has a height of 144 meters and a
volume of 2,226,450 m3, what are the dimensions of the base?
x = ____________________
Example 3
Find the volume of the right cone. V ≈ _________________________
Example 4
Find the volume of the solid shown.
V = _____________________
Example 5
Unit 4: Surface Area and Volume of Solids
Lesson 4.6 Surface Area and Volume of Spheres Lesson 12.6 from textbook
Objectives: • Identify the different parts of the sphere: center, radius, diameter, and great circle.
• Use the given formulas to find the surface area and volume of a sphere.
• Use the given volume and surface area to find the unknown measures of the sphere.
• Solve problems involving unit conversion for situations involving volumes within the same
measurement system.
SPHERE GREAT CIRCLES and
HEMISPHERES
Surface Area of a Sphere Theorem
S = __________________________
Example 1 Example 2
Find the surface area of the sphere. Find the radius of the sphere if the surface area is 228 cm2.
r = ___________________
S = _________________________
Example 3
Find the surface area of the composite solid.
S = ____________________
Volume of a Sphere Theorem Example 4
Find the volume of the sphere.
V = _____________________
V = ______________________________
Example 5
Find the volume of the composite solid.
V = ______________________________
Example 6