11.6 Volumes of Pyramids Essential Question How can you find te volume of a pyramid? Finding te Volume of a Pyramid Work wit a partner. Te pyramid and te prism ave te same eigt and te same square base. Wen te pyramid is filled wit sand and poured into te prism, it takes tree pyramids to fill te prism. OOIG FOR STRUCTURE To be proficient in mat, you need to look closely to discern a pattern or structure. Use tis information to write a formula for te volume V of a pyramid. Finding te Volume of a Pyramid Work wit a partner. Use te formula you wrote in Exploration 1 to find te volume of te exagonal pyramid. in. 2 in. Communicate Your Answer. How can you find te volume of a pyramid? 4. In Section 11.7, you will study volumes of cones. How do you tink you could use a metod similar to te one presented in Exploration 1 to write a formula for te volume of a cone? Explain your reasoning. Section 11.6 Volumes of Pyramids 65
11.6 esson Wat You Will earn Core Vocabulary Previous pyramid composite solid Find volumes of pyramids. Use volumes of pyramids. Finding Volumes of Pyramids Consider a triangular prism wit parallel, congruent bases J and P. You can divide tis triangular prism into tree triangular pyramids. Pyramid Q P Pyramid R J P prism J pyramid 1 pyramid 2 J P pyramid You can combine triangular pyramids 1 and 2 to form a pyramid wit a base tat is a parallelogram, as sown at te left. ame tis pyramid Q. Similarly, you can combine triangular pyramids 1 and to form pyramid R wit a base tat is a parallelogram. In pyramid Q, diagonal divides J into two congruent triangles, so te bases of triangular pyramids 1 and 2 are congruent. Similarly, you can divide any cross section parallel to J into two congruent triangles tat are te cross sections of triangular pyramids 1 and 2. By Cavalieri s Principle, triangular pyramids 1 and 2 ave te same volume. Similarly, using pyramid R, you can sow tat triangular pyramids 1 and ave te same volume. By te Transitive Property of Equality, triangular pyramids 2 and ave te same volume. Te volume of eac pyramid must be one-tird te volume of te prism, or 1 B. You can generalize tis formula to say tat te volume of any pyramid wit any base is equal to 1 te volume of a prism wit te same base and eigt because you can divide any polygon into triangles and any pyramid into triangular pyramids. Core Concept Volume of a Pyramid Te volume V of a pyramid is 1 B were B is te area of te base and is te eigt. B B Find te volume of te pyramid. Finding te Volume of a Pyramid SOUTIO 66 Capter 11 Circumference, Area, and Volume 1 B Formula for volume of a pyramid = 1 ( 1 2 4 6 ) (9) Substitute. = 6 Simplify. Te volume is 6 cubic meters. 9 m 4 m
onitoring Progress Find te volume of te pyramid. Help in Englis and Spanis at BigIdeasat.com 1. 12 cm 2. 20 cm 10 cm 12 cm Using Volumes of Pyramids Using te Volume of a Pyramid Originally, afre s Pyramid ad a eigt of about 144 meters and a volume of about 2,218,800 cubic meters. Find te side lengt of te square base. afre s Pyramid, Egypt SOUTIO 1 B Formula for volume of a pyramid 2,218,800 1 x2 (144) Substitute. 6,656,400 144x 2 ultiply eac side by. 46,225 x 2 Divide eac side by 144. 215 x Find te positive square root. Originally, te side lengt of te square base was about 215 meters. Using te Volume of a Pyramid Find te eigt of te triangular pyramid. V = 14 ft V = 24 m m SOUTIO Te area of te base is B = 1 2 ()(4) = 6 ft2 and te volume is V = 14 ft. 1 B Formula for volume of a pyramid 14 = 1 (6) Substitute. 7 = Solve for. Te eigt is 7 feet. onitoring Progress Help in Englis and Spanis at BigIdeasat.com. Te volume of a square pyramid is 75 cubic meters and te eigt is 9 meters. Find te side lengt of te square base. 4. Find te eigt of te triangular pyramid at te left. 4 ft ft Section 11.6 Volumes of Pyramids 67
Finding te Volume of a Similar Solid Pyramid A and pyramid B are similar. Find te volume of pyramid B. 8 m Pyramid A Pyramid B V = 9 SOUTIO Heigt of pyramid B Te scale factor is k = Heigt of pyramid A = 6 8 = 4. Use te scale factor to find te volume of pyramid B. Volume of pyramid B Volume of pyramid A = k Te ratio of te volumes is k. 4) Volume of pyramid B = 96 ( Substitute. Volume of pyramid B = 40.5 Solve for volume of pyramid B. Te volume of pyramid B is 40.5 cubic meters. onitoring Progress Help in Englis and Spanis at BigIdeasat.com 5. Pyramid C and pyramid D are similar. Find te volume of pyramid D. Pyramid C Pyramid D V = 24 m m 9 m Finding te Volume of a Composite Solid Find te volume of te composite solid. SOUTIO Volume of solid = Volume of cube + Volume of pyramid = s + 1 B Write formulas. = 6 + 1 (6)2 6 Substitute. = 216 + 72 Simplify. ft = 288 Add. Te volume is 288 cubic meters. 5 ft onitoring Progress Help in Englis and Spanis at BigIdeasat.com 4 ft 8 ft 6. Find te volume of te composite solid. 68 Capter 11 Circumference, Area, and Volume
11.6 Exercises Dynamic Solutions available at BigIdeasat.com Vocabulary and Core Concept Ceck 1. VOCABUARY Explain te difference between a triangular prism and a triangular pyramid. 2. REASOIG A square pyramid and a cube ave te same base and eigt. Compare te volume of te square pyramid to te volume of te cube. onitoring Progress and odeling wit atematics In Exercises and 4, find te volume of te pyramid. (See Example 1.). 7 m 12 m 1 4. in. In Exercises 5 8, find te indicated measure. (See Example 2.) 4 in. in. 5. A pyramid wit a square base as a volume of 120 cubic meters and a eigt of 10 meters. Find te side lengt of te square base. 10. OPE-EDED Give an example of a pyramid and a prism tat ave te same base and te same volume. Explain your reasoning. In Exercises 11 14, find te eigt of te pyramid. (See Example.) 11. Volume = 15 ft 12. Volume = 224 in. ft ft 8 in. 1. Volume = 198 yd 14. Volume = 92 cm 6. A pyramid wit a square base as a volume of 912 cubic feet and a eigt of 19 feet. Find te side lengt of te square base. 7. A pyramid wit a rectangular base as a volume of 480 cubic inces and a eigt of 10 inces. Te widt of te rectangular base is 9 inces. Find te lengt of te rectangular base. 8. A pyramid wit a rectangular base as a volume of 105 cubic centimeters and a eigt of 15 centimeters. Te lengt of te rectangular base is 7 centimeters. Find te widt of te rectangular base. 11 yd 9 yd 14 cm 7 cm In Exercises 15 and 16, te pyramids are similar. Find te volume of pyramid B. (See Example 4.) 15. 12 ft Pyramid A Pyramid B ft 9. ERROR AAYSIS Describe and correct te error in finding te volume of te pyramid. 5 ft V = 1 (6)(5) = 1 (0) 16. V = 256 ft Pyramid A Pyramid B 6 ft = 10 ft in. V = 10 in. 6 in. Section 11.6 Volumes of Pyramids 69
In Exercises 17 20, find te volume of te composite solid. (See Example 5.) 17. 18. in. 7 cm 2. CRITICA THIIG Find te volume of te regular pentagonal pyramid. Round your answer to te nearest undredt. In te diagram, m ABC = 5. A 10 cm 2 in. 6 in. 4 in. 19. 20. 5 cm 5 cm 8 cm 12 cm 9 cm C ft B 24. THOUGHT PROVOIG A frustum of a pyramid is te part of te pyramid tat lies between te base and a plane parallel to te base, as sown. Write a formula for te volume of te frustum of a square pyramid in terms of a, b, and. (Hint: Consider te missing top of te pyramid and use similar triangles.) b b 21. ABSTRACT REASOIG A pyramid as a eigt of 8 feet and a square base wit a side lengt of 6 feet. a. How does te volume of te pyramid cange wen te base stays te same and te eigt is doubled? b. How does te volume of te pyramid cange wen te eigt stays te same and te side lengt of te base is doubled? c. Are your answers to parts (a) and (b) true for any square pyramid? Explain your reasoning. 22. HOW DO YOU SEE IT? Te cube sown is formed by tree pyramids, eac wit te same square base and te same eigt. How could you use tis to verify te formula for te volume of a pyramid? a 25. ODEIG WITH ATHEATICS autical deck prisms were used as a safe way to illuminate decks on sips. Te deck prism sown ere is composed of te following tree solids: a regular exagonal prism wit an edge lengt of.5 inces and a eigt of 1.5 inces, a regular exagonal prism wit an edge lengt of.25 inces and a eigt of 0.25 inc, and a regular exagonal pyramid wit an edge lengt of inces and a eigt of inces. Find te volume of te deck prism. a aintaining atematical Proficiency Find te value of x. Round your answer to te nearest tent. (Section 9.4 and Section 9.5) 26. 9 27. 57 15 Reviewing wat you learned in previous grades and lessons 28. 29. x 64 x 7 5 x x 10 0 640 Capter 11 Circumference, Area, and Volume