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