Surface Area of Pyramids 393

Surface Area 7-8 of Pyramids

MAIN IDEA MUSEUMS The Rock and Roll Hall Find the lateral and total surface areas of of Fame and Museum opened in pyramids. Cleveland, Ohio, in 1995.

New Vocabulary 1. Not including the base, how many faces does this pyramid regular pyramid have? What shape are they? slant height 2. How could you find the total area Math Online of the glass used for the building? glencoe.com • Extra Examples • Personal Tutor • Self-Check Quiz A regular pyramid is a pyramid with a base that is a regular polygon. The lateral faces of a regular pyramid are congruent isosceles triangles. At the top of the pyramid, these triangles meet at a common point called the vertex. The altitude or height of each lateral face is called the slant height of the pyramid.

Model of Regular Pyramid Net of Regular Pyramid

vertex base lateral face lateral face slant height

base slant height side length s of regular polygon

To find the lateral area L of a regular pyramid, look at its net. The lateral area of a pyramid is the sum of the areas of its lateral faces, which are all triangles.

The net of a square pyramid is a square and four triangles as shown above.

1 L = 4 _ s Area of the lateral faces (2 ) 1 L = _ (4s) Commutative Property of Multiplication 2 1 L = _ P The perimeter of the base P is 4s. 2

The total surface area of a regular pyramid is the lateral surface area plus the area of the base.

Lesson 7-8 Surface Area of Pyramids 393

393_396_C07_L08_874050.indd 393 9/21/07 9:12:16 AM Gr8 MS Math SE ©09 - 874050 Lateral Surface Area of a Pyramid Key Concept

Words The lateral surface area L of a Model regular pyramid is half the perimeter P of the base times the slant height . 1 Symbols L = _ P 2 Total Surface Area of a Pyramid Key Concept

Words The total surface area S of a Model C07-098A-874050.ai regular pyramid is the lateral area L plus the area of the base B. 1 Symbols S = L + B or S = _ P + B 2

Surface Areas of a PyramidC07-154A-874050.ai

1 Find the lateral and total surface areas of 10 ft 12 ft the triangular pyramid. 1 L = _ P S = L + B 2 8.7 ft 10 ft 10 ft 1 1 L = _ · 30 · 12 S = 180 + 43.5 B = _ · 10 · 8.7 2 2 L = 180 S = 223.5 The lateral and total surface areas are 180 and 223.5 square feet. C07-99A-877850.ai

a. Find the lateral and total surface areas of a pyramid with a slant height of 18 meters and a square base with 11-meter sides.

2 ARCHITECTURE Use the information at the left to find the lateral surface area of the Pyramid of the Sun if it has a slant height of 132.5 meters. Real-World Link 1 The Pyramid of the L = _ P Lateral surface area of a pyramid Sun in Teotihuacán, 2 Mexico, was built in 1 L = _ · 894 · 132.5 P = 223.5(4) or 894 and = 132.5 the second century, 2 A.D. It is about 71 meters tall, and L = 59,227.5 Simplify. its square base has side lengths The lateral area of the pyramid is 59,227.5 square meters. of 223.5 meters.

b. AWARDS A music award is a square pyramid with a 6-inch-long base and a 13-inch slant height. Find the award’s total surface area.

394 Chapter 7 Measurement: Area and Volume

394_C07_L08_874050.inddGr8 MS Math 394SE ©09 - 874050 7/23/09 2:32:33 PM Example 1 Find the lateral and total surface areas of each regular pyramid. Round to the (p. 394) nearest tenth if necessary.

1. 2. 12 m 15 m 6 ft

10.2 m 4 ft 12 m 12 m 4 ft

Example 2 3. EVENTS The Pyramid Arena in Memphis is a regular square pyramid. Each (p. 394) face of the arena has a base of 600 feet and a height of about 477 feet. Find the lateral surface area of the pyramid.

Find the lateral and total surface area of each regular pyramid. Round to the HOMEWORK HELP nearest tenth if necessary. For See Exercises Examples 4. 5. 6. 6 m 8.3 m 3.5 in. 5 ft 4–9 1 10, 11 2 6 m 1 6 m 2 in. ft3 2 in. 1 2 2 ft3 5.2 m

7. 7.8 mm 8. 18 cm 9. 32 ft 9 mm

7.8 mm 15 cm 24 ft 9 mm 9 mm 15 cm 24 ft

10. ARCHITECTURE The Transamerica Pyramid in San Francisco is shaped like a square pyramid. It has a slant height of 856.1 feet and each side of its base is 145 feet. Find the lateral area of the building.

11. ROOFS A pyramid-shaped roof has a slant height of 16 feet and its square base is 40 feet wide. How much roofing material is needed to cover the roof?

12. A square pyramid has a lateral area of 107.25 square centimeters and a slant height of 8.25 centimeters. Find the length of each side of its base.

13. ARCHAEOLOGY The Pyramid of Khafre ft in Egypt stands 471 feet tall. The sides 471 ft of its square base are 705 feet in length. Find the lateral surface area of 705 ft the Pyramid of Khafre. (Hint: Use the PRACTICE 705 ft EXTRA Pythagorean Theorem to find the See pages 688, 706. pyramid’s slant height .)

Lesson 7-8 Surface Area of Pyramids 395

393_396_C07_L08_874050.indd 395 9/21/07 9:12:22 AM Gr8 MS Math SE ©09 - 874050 H.O.T. Problems CHALLENGE For Exercises 14–16, use the drawings of the figure shown. The total height of the figure is 20 inches. 4JEF7JFXPG1ZSBNJE 14. Find the height h of the pyramid. 15. Use the height of the pyramid to find the slant height, l. 5 in. 16 in. 16 in. 16. Which has a greater surface area, the prism or the pyramid? Explain 8 in. 8 in. your reasoning.

17. OPEN ENDED A pyramid has a base that is 3 inches square and a slant height of 4 inches. A rectangular prism has the same surface area. Give possible side lengths of the prism. 18. WRITING IN MATH Explain how you can use the height of a pyramid to find the slant height.

19. Which is the best estimate for the 20. The net of a paperweight is shown surface area of the pyramid? below. Which is closest to the lateral surface area of the paperweight? 16 ft

7 cm

13.4 ft 13.4 ft

A 107 ft 2 9.1 cm B 180 ft 2 C 429 ft 2 F 32 cm 2 H 127 cm 2 D 608 ft 2 G 49 cm 2 J 176 cm 2

21. PACKAGING Find the surface area of a can that has a diameter of 3 inches and a height of 5 inches. (Lesson 7-7)

22. MOUNTAINS A student is creating a clay model of a mountain shaped like a cone. If the mountain is 4 feet tall and the radius of the base is 2 feet, what is the volume of clay needed to make the mountain? Round to the nearest tenth if necessary. (Lesson 7-6)

PREREQUISITE SKILL Solve each proportion. (Lesson 4-2) 16 12 3 x a 7 10 30 23. _ = _ 24. _ = _ 25. _ = _ 26. _ = _ n 40 5 8 13 39 26 w

396 Chapter 7 Measurement: Area and Volume