How do you find the volume of a triangular prism or a triangular pyramid?

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ESSENTIAL QUESTION

Finding the Volume of a Triangular PrismThe volume V of a prism is the area of its base B times its height h, or V = Bh.

Find the volume of the triangular prism.

Find the area of the triangular base.

Find the volume of the triangular prism.

The volume of the triangular prism is 84 cubic feet.

Reflect1. What If? Suppose you know the volume V and base area B

of a triangular prism. How could you find the height of the prism?

EXAMPLEXAMPLE 1

STEP 1

A = 1 _ 2

bh

A = 1 _ 2

(8 · 3)

A = 12 f t 2

Write the formula.

Substitute 8 for the length of the triangular base and 3 for the height of the triangle.

STEP 2

V = Bh

V = 12 · 7

V = 84 f t 3

Write the formula.

Substitute the value from Step 1 for B and 7 for the height of the prism.

L E S S O N

10.2Volume of Triangular Prisms and Pyramids

2. Find the volume of a garden seat in the shape of a triangular prism

with a height of 30 inches and a base area of 72 i n 2 .

YOUR TURN

Math TalkMathematical Processes

Equations, expressions, and relationships—7.9.A Solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. Also 7.8.B

7.9.A

Explain the difference between the variables

b and B in the formulas A = 1 _

2 bh and V = Bh.

323Lesson 10.2

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5. The volume of a triangular pyramid is 13.5 m 3 . What is the volume of a

triangular prism with a congruent base and the same height? Explain.

YOUR TURN

Exploring the Volume of a Triangular PyramidPreviously you explored the volumes of rectangular prisms and pyramids. Now

you will repeat the same activity with triangular pyramids and prisms that have

the same height and congruent bases.

Make three-dimensional models. Make

larger versions of the nets shown. Make

sure the bases and heights in each net are

the same size. Fold each net, and tape it

together to form an open prism or pyramid.

Fill the pyramid with beans. Make sure that the beans are level with the

opening of the pyramid. How many pyramids full of beans do you think

it will take to fill the prism?

Pour the beans into the prism. Repeat until the prism is full.

Was your conjecture supported?

Write a fraction that compares the volume of the pyramid to the

volume of the prism.

Reflect3. Analyze Relationships Does it appear that the relationship

between the volume of triangular pyramids and prisms is the

same as that for rectangular pyramids and prisms?

4. Draw Conclusions Write a formula for the volume of a

triangular pyramid with a base area of B and a height of h.

STEP 1

STEP 2

STEP 3

EXPLORE ACTIVITY

7.8.B

Unit 5324

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My Notes

2.5 ft

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Solving Volume ProblemsAs you solve volume problems, you will use the volume formulas you have

learned. You will also need to use the formula for the area of a triangle: A = 1 _ 2 bh.

Mr. Martinez is building wooden shapes for a sculpture

in the park. His plans show a triangular pyramid and a

triangular prism, and each shape is 5 feet high. The base of

each shape is a triangle with a base of 2.5 feet and a height

of 2 feet. How much greater than the volume of the

pyramid is the volume of the prism?

The triangular base is the same for both shapes.

Find the area of the base B.

The area of the base B for both shapes is 2.5 f t 2 .

Find the volume of the prism.

V = Bh

V = (2.5) × 5 = 12.5

The volume of the prism is 12.5 f t 3 .

Find the volume of the pyramid.

Volume of pyramid = 1 _ 3

· volume of prism

= 1 _ 3

· 12.5 ≈ 4.2

The volume of the pyramid is approximately 4.2 f t 3 .

Compare the volumes.

Volume of prism - volume of pyramid = 12.5 - 4.2 = 8.3

The volume of the prism is 8.3 f t 3 greater than that of the pyramid.

EXAMPLEXAMPLE 2

STEP 1

A = 1 __ 2

bh

A = 1 __ 2

(2.5)(2) = 2.5 Substitute the values for b and h of the triangle.

The area A of the triangle is the same as B in V = Bh.

STEP 2

Substitute the value for B and h of the prism.

STEP 3

Substitute and calculate.

STEP 4

Use the formula.

6. How much greater is the volume of a triangular prism with base area

of 14 c m 2 and height of 4.8 cm than the volume of a triangular pyramid

with the same height and base area?

YOUR TURN

7.9.A

325Lesson 10.2

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3 in.

16 in.

3 in.

5 ft9 ft

9 ft

6 yd6 yd

6 yd

10 in.7 in.

5 in.

Guided Practice

1. Find the volume of the triangular prism. (Example 1)

Find the area of the base of the prism.

Use the equation A = bh.

A = ( ) ( ) = in 2

Find the volume of the prism. Use the equation V = Bh.

V = ( ) ( ) = in 3

2. A triangular pyramid and a triangular prism have congruent bases and

the same height. The triangular pyramid has a volume of 90 m 3 . Find

the volume of the prism. (Explore Activity)

The volume of the prism is because the volume of the prism

is times the volume of the pyramid.

3. In Exercise 2, how much greater is the volume of the prism than the

volume of the pyramid? (Example 2)

4. 5. 6.

Find the volume of each figure.

7. A pyramid has a base that is a triangle. The length of the base of the

triangle is 5 meters, and the height of the triangle is 12 meters. The height

of the pyramid is 10 meters. How would you explain to a friend how to

find the volume of the pyramid?

ESSENTIAL QUESTION CHECK-IN??

Unit 5326

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6 in.

25 in.7 in.

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Name Class Date

Independent Practice10.2

8. A trap for insects is in the shape of a triangular prism. The area of the

base is 3.5 in 2 and the height of the prism is 5 in. What is the volume

of this trap?

9. Arletta built a cardboard ramp for her little brothers’ toy cars.

Identify the shape of the ramp. Then find its volume.

10. Represent Real-World Problems Sandy builds this shape of four

congruent triangles using clay and toothpicks. The area of each triangle is

17.6 cm 2 , and the height of the shape is 5.2 cm. What three-dimensional

figure does the shape Sandy built resemble? If this were a solid shape,

what would be its volume? Round your answer to the nearest tenth.

11. Draw Conclusions Would tripling the height of a triangular prism triple

its volume? Explain.

12. The Jacksons went camping in a state park.

One of the tents they took is shown. What is the

volume of the tent?

13. Shawntelle is solving a problem involving a

triangular pyramid. You hear her say that “bee” is equal to

24 inches. How can you tell if she is talking about the

base area B of the pyramid or about the base b of the triangle?

14. Alex made a sketch for a homemade soccer goal he plans to build. The

goal will be in the shape of a triangular prism. The legs of the right

triangles at the sides of his goal measure 4 ft and 8 ft, and the opening

along the front is 24 ft. How much space is contained within this goal?

7.9.A, 7.8.B

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Work Area

6 cm8 cm

10 cm

2.5 cm

2 cm

4 cm

5 cm

15. A plastic puzzle in the shape of a triangular prism has bases that are

equilateral triangles with side lengths of 4 inches and a height of 3.5

inches. The height of the prism is 5 inches. Find the volume of the prism.

16. Persevere in Problem Solving Lynette’s

grandmother has a metal doorstop with

the dimensions shown. Find the volume of

the metal in the doorstop. The metal in the

doorstop has a mass of about 8.6 grams

per cubic centimeter. Find the mass of the

doorstop.

17. Make a Conjecture Don and Kayla each draw a triangular pyramid

that has a volume of 100 cm 3 . They do not draw identical shapes.

Give a set of possible dimensions for each pyramid.

18. Multistep Don’s favorite cheese snack

comes in a box of six pieces. Each piece

of cheese has the shape of a triangular

prism that is 2 cm high. The triangular

base of the prism has a height of 5 cm

and a base of 4 cm. Find the volume of

cheese in the box.

19. Analyze Relationships What effect would doubling all the dimensions

of a triangular pyramid have on the volume of the pyramid? Explain your

reasoning.

FOCUS ON HIGHER ORDER THINKING

Unit 5328

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