? LESSON 10.2 ESSENTIAL QUESTION How do you find the volume of a triangular prism or a triangular pyramid? Finding the Volume of a Triangular Prism The volume V of a prism is the area of its base B times its height h, or V = Bh. EXAMPLE 1 Volume of Triangular Prisms and Pyramids 7.9.A 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 Math On the Spot Find the volume of the triangular prism. STEP 1 STEP 2 Find the area of the triangular base. A = 1_ 2 bh A = 1_ (8 3) 2 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. Find the volume of the triangular prism. 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. The volume of the triangular prism is 84 cubic feet. Reflect 1. 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? 3 ft 7 ft 8 ft Math Talk Mathematical Processes Explain the difference between the variables b and B in the formulas A = 1_ bh and V = Bh. 2 YOUR TURN 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. Online Assessment and Intervention Lesson 10.2 323
EXPLORE ACTIVITY 7.8.B Exploring the Volume of a Triangular Pyramid Previously 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. STEP 1 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. STEP 2 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? STEP 3 Write a fraction that compares the volume of the pyramid to the volume of the prism. 3. 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. YOUR TURN Online Assessment and Intervention 324 Unit 5 5. The volume of a triangular pyramid is 13.5 m3. What is the volume of a triangular prism with a congruent base and the same height? Explain. Reflect
Solving Volume Problems As 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. EXAMPLE 2 7.9.A Math On the Spot 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? 2. 2 ft My Notes STEP 1 The triangular base is the same for both shapes. Find the area of the base B. A = 1 2 bh The area A of the triangle is the same as B in V = Bh. A = 1 (2.5)(2) = 2.5 Substitute the values for b 2 and h of the triangle. The area of the base B for both shapes is 2.5 f t 2. 2. 2 ft STEP 2 Find the volume of the prism. V = Bh V = (2.5) 5 = 12.5 Use the formula. Substitute the value for B and h of the prism. The volume of the prism is 12.5 f t 3. STEP 3 Find the volume of the pyramid. STEP 4 YOUR TURN Volume of pyramid = 1_ 3 volume of prism = 1_ 3 12.5 4.2 Substitute and calculate. 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. 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? Online Assessment and Intervention Lesson 10.2 325
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. 3 in. 3 in. 16 in. 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) Find the volume of each figure. 4. 5. 6. 6 yd 9 ft 6 yd 9 ft 6 yd 7 in. 10 in. 5 in.? ESSENTIAL QUESTION CHECK-IN 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? 326 Unit 5
Name Class Date 10.2 Independent Practice 7.9.A, 7.8.B Online Assessment and Intervention 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. 6 in. 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. 25 in. 7 in. 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? 3. 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? 6 ft 4. Lesson 10.2 327
FOCUS ON HIGHER ORDER THINKING Work Area 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. 10 cm 2.5 cm 8 cm 6 cm 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. 4 cm 2 cm 5 cm 19. Analyze Relationships What effect would doubling all the dimensions of a triangular pyramid have on the volume of the pyramid? Explain your reasoning. 328 Unit 5