Base

Lateral face

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Lateral and Total Surface Area of a PrismThe lateral faces of a prism are parallelograms that connect the bases. Each face

that is not a base is a lateral face. The sum of the areas of all the lateral faces is

the lateral area of the prism. The surface area is the sum of areas of all of the

surfaces of a figure expressed in square units. The total surface area of a prism

can be found by finding the sum of the lateral area and the area of the bases.

You can find the lateral area and total surface area of a prism by using a net.

A net of a rectangular prism is shown.

Use the net to find the lateral area and the

total surface area of the prism. Each square

represents one square inch. The blue regions

are the bases, and the green regions are the

lateral faces.

Find the lateral area of the

rectangular prism. There are

two 2 in. by 5 in. rectangles, and two 2 in. by 6 in. rectangles.

2 · (2 · 5) = 20 in 2

2 · (2 · 6) = 24 in 2 .

The lateral area is 20 + 24 = 44 in 2 .

Find the total surface area of the rectangular prism.

Each base is 5 inches by 6 inches. 2 · (5 · 6) = 60 in 2 .

The total surface area is 44 + 60 = 104 in 2 .

EXAMPLEXAMPLE 1

STEP 1

STEP 2

ESSENTIAL QUESTIONHow do you find the lateral and total surface area of rectangular and triangular prisms or pyramids?

L E S S O N

10.3Lateral and Total Surface Area

Math TalkMathematical Processes

7.9.A

Equations, expressions, and relationships—7.9.D Solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape’s net.

Explain how the total surface area of a prism differs from

the lateral area.

329Lesson 10.3

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16 in. 16 in.

17 in.17 in.

16 in.

16 in.

4 cm

4 cm

3 cm

4 cm 5 cm

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1. Use the net to find the lateral area and

the total surface area of the triangular

prism described by the net.

YOUR TURN

Lateral and Total Surface Area of a PyramidIn a rectangular pyramid, the base is a rectangle, and the lateral faces are

triangles. The lateral area and the total surface area are defined in the same

way as they are for a prism.

Use the net of this rectangular pyramid to find the lateral area and the

total surface area of this pyramid.

Find the lateral area of the rectangular pyramid.

There are four triangles with base 16 in. and height 17 in.

The lateral area is 4 × 1 _ 2 (16)(17) = 544 in 2 .

Find the total surface area of the rectangular pyramid.

The area of the base is 16 × 16 = 256 in 2 .

The total surface area is 544 + 256 = 800 in 2 .

Reflect2. How many surfaces does a triangular pyramid have? What shape are they?

EXAMPLE 2

STEP 1

STEP 2

7.9.D

Unit 5330

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20 in.

16 in. 16 in.

20 in.

2.6 ft3 ft

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2.5 ft2.5 ft

3.5 ft

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3. The base and all three faces of a triangular pyramid

are equilateral triangles with side lengths of 3 ft.

The height of each triangle is 2.6 ft. Use the net to

find the lateral area and the total surface area of

the triangular pyramid.

4. Use a net to find the lateral area and the

total surface area of the pyramid.

YOUR TURN

Solving Area ProblemsSeveral students have volunteered to design and build structures for the

Lincoln Middle School art exhibit.

Shoshanna’s team plans to build stands to display sculptures. Each stand

will be in the shape of a rectangular prism. The prism will have a square

base with side lengths of 2 1 _ 2

feet, and it will be 3 1 _ 2

feet high. The team

plans to cover the stands with metallic foil that costs $0.22 per square foot.

How much money will the team save on each stand if they cover only the

lateral area instead of the total surface area?

Make a net of the rectangular prism.

Find the lateral and total surface

areas of the prism.

The four lateral faces are 3 1 _ 2 feet by

2 1 _ 2 feet rectangles. The two bases

are 2 1 _ 2 feet by 2 1 _

2 feet squares.

Lateral area: 4 ·  (3 1 _ 2 · 2 1 _

2 ) = 35 square feet

Total surface area: 2 · ( 2 1 _ 2 · 2 1 _

2 ) + 35 = 47 1 _

2 square feet

Find and compare the prices.

Cost for total surface area: $0.22 ( 47 1 _ 2 ) = $10.45

Cost for lateral area: $0.22(35) = $7.70

The team will save $10.45 - $7.70, or $2.75, for each stand by

covering only the lateral area.

EXAMPLEXAMPLE 3

STEP 1

STEP 2

STEP 3

7.9.D

Add the area of the bases to the lateral area.

331Lesson 10.3

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4 cm4.3 cm

3 cm

4.3 cm6 cm

10 cm10 cm

12 cm

3 ft

6.1 ft

5. Kwame’s team will make two triangular pyramids

to decorate the entrance to the exhibit. They will be

wrapped in the same metallic foil. Each base is an

equilateral triangle. If the base has an area of about

3.9 square feet, how much will the team save altogether

by covering only the lateral area of the two pyramids?

The foil costs $0.22 per square foot.

YOUR TURN

Guided Practice

A three-dimensional figure is shown sitting on a base. (Example 1)

1. The figure has a total of rectangular faces.

2. Of the total number of faces, are lateral faces.

3. The figure is a .

4. Sketch a net of the figure.

5. The lateral area of the prism is .

6. The total surface area is .

A triangular prism is shown. (Example 2)

7. Identify the number and type of faces of the prism.

8. Find the lateral area of the prism.

9. Find the total surface area of the prism.

10. Use a net to find the total surface area of the pyramid. Then find the

cost of wrapping the pyramid completely in gold foil that costs $0.05 per

square centimeter. (Examples 2 and 3)

11. How do you find the lateral and total surface area of a triangular pyramid?

ESSENTIAL QUESTION CHECK-IN??

Unit 5332

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3 in.

15 in.

3.6 in.

4 in.

12 in.

8 in.2 in.

8 in.6 in.

5 in.

20 cm 15 cm

9 cm

3 ft

2 ft

2 ft

2.5 ft

2.5 ft

4 ft

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Name Class Date

12. Use a net to find the lateral area and the total

surface area of the cereal box.

Lateral area:

Total surface area:

13. Describe a net for the shipping carton shown.

14. A shipping carton is in the shape of a triangular prism.

Use a net to find the lateral area and the total surface

area of the carton.

Lateral area:

Total surface area:

15. Victor wrapped this gift box with adhesive paper (with no overlaps).

How much paper did he use?

16. Vocabulary Name a three-dimensional shape that has four triangular

faces and one rectangular face.

17. Cindi wants to cover the top and sides of this box with glass tiles that

are 1 cm square. How many tiles will she need?

18. A glass paperweight has the shape of a triangular prism. The bases

are equilateral triangles with side lengths of 4 inches and heights of

3.5 inches. The height of the prism is 5 inches. Find the lateral

area and the total surface area of the paperweight.

19. The doghouse shown has a floor, but no windows. Find the total

surface area of the doghouse (including the door).

20. Describe the simplest way to find the total surface area of a cube.

Independent Practice10.37.9.D

333Lesson 10.3

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Work Area

21. Communicate Mathematical Ideas Describe how you approach

a problem involving lateral area and total surface area. What do you do

first? In what ways can you use the figure that is given with a problem?

What are some shortcuts that you might use when you are calculating

these areas?

22. Persevere in Problem Solving A pedestal in a craft store is in the shape

of a triangular prism. The bases are right triangles with side lengths of

12 cm, 16 cm, and 20 cm. The store owner used 192 cm 2 of burlap cloth

to cover the lateral area of the pedestal. Find the height of the pedestal.

23. Communicate Mathematical Ideas The base of Prism A has an area of

80 ft 2 , and the base of Prism B has an area of 80 ft 2 . The height of Prism A

is the same as the height of Prism B. Is the base of Prism A congruent to

the base of Prism B? Explain.

24. Critique Reasoning A triangular pyramid is made of 4 equilateral

triangles. The sides of the triangles measure 5 m, and the height of each

triangle is 4.3 m. A rectangular prism has a height of 4.3 m and a square

base that is 5 m on each side. Susan says that the total surface area of

the prism is more than twice the total surface area of the pyramid. Is she

correct? Explain.

FOCUS ON HIGHER ORDER THINKING

Unit 5334

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