MUSEUMS The Rock and Roll Hall

of Fame and Museum opened in

Cleveland, Ohio, in 1995.

1. Not including the base, how

many faces does this pyramid

have? What shape are they?

2. How could you find the total area

of the glass used for the building?

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.

Net of Regular PyramidModel of Regular Pyramid

vertex

slant height

slant height

lateral face lateral face

side length s ofregular polygon

base

base

�

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.

L = 4 ( 1 _ 2 s�) Area of the lateral faces

L = 1 _ 2 (4s)� Commutative Property of Multiplication

L = 1 _ 2 P� The perimeter of the base P is 4s.

The total surface area of a regular pyramid is the lateral surface area

plus the area of the base.

Surface Area of Pyramids7-8

MAIN IDEAFind the lateral and total surface areas of pyramids.

New Vocabularyregular pyramidslant height

Math Online

glencoe.com• Extra Examples• Personal Tutor• Self-Check Quiz

Lesson 7-8 Surface Area of Pyramids 393

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Surface Areas of a Pyramid

1 Find the lateral and total surface areas of 12 ft10 ft

10 ft10 ft8.7 ft

C07-99A-877850.ai

the triangular pyramid.

L = 1 _ 2 P� S = L + B

L = 1 _ 2 · 30 · 12 S = 180 + 43.5 B = 1 _

2 · 10 · 8.7

L = 180 S = 223.5

The lateral and total surface areas are 180 and 223.5 square feet.

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 of132.5 meters.

L = 1 _ 2 P� Lateral surface area of a pyramid

L = 1 _ 2 · 894 · 132.5 P = 223.5(4) or 894 and � = 132.5

L = 59,227.5 Simplify.

The lateral area of the pyramid is 59,227.5 square 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.

Real-World LinkThe Pyramid of the Sun in Teotihuacán, Mexico, was built in the second century, A.D. It is about 71 meters tall, and its square base has side lengths of 223.5 meters.

Words The lateral surface area L of a regular pyramid is half the perimeter P of the base times the slant height �.

Model

Symbols L = 1 _ 2 P�

Key ConceptLateral Surface Area of a Pyramid

Words The total surface area S of a regular pyramid is the lateral area L plus the area of the base B.

Model

Symbols S = L + B or S = 1 _ 2 P� + B

Key ConceptTotal Surface Area of a Pyramid

C07-154A-874050.ai

C07-098A-874050.ai

394 Chapter 7 Measurement: Area and Volume

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705 ft

705 ft

� ft

471 ft

Example 1(p. 394)

Find the lateral and total surface areas of each regular pyramid. Round to the

nearest tenth if necessary.

1.6 ft

4 ft4 ft

2. 12 m 15 m

12 m10.2 m

12 m

Example 2(p. 394)

3. EVENTS The Pyramid Arena in Memphis is a regular square pyramid. Each

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

nearest tenth if necessary.

4.

2 in.2 in.

3.5 in. 5.

ft3 12

ft3 12

5 ft 6. 8.3 m

6 m 6 m

5.2 m

6 m

7.

9 mm 9 mm

9 mm 7.8 mm

7.8 mm

8. 18 cm

15 cm 15 cm

9. 32 ft

24 ft 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

in Egypt stands 471 feet tall. The sides

of its square base are 705 feet in

length. Find the lateral surface area of

the Pyramid of Khafre. (Hint: Use the

Pythagorean Theorem to find the

pyramid’s slant height �.)

For Exercises

See Examples

4–910, 11

12

HOMEWORK HELPHELP

See pages 688, 706.EXTRA PRACTICEPRACTICE

Lesson 7-8 Surface Area of Pyramids 395

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�

8 in. 8 in.

16 in.16 in.5 in.

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)

23. 16 _ n = 12

_ 40

24. 3 _ 5 = x _

8 25. a _

13 = 7 _

39 26. 10

_ 26

= 30 _ w

CHALLENGE For Exercises 14–16, use the drawings of the figure shown.

The total height of the figure is 20 inches.

14. Find the height h of the pyramid.

15. Use the height of the pyramid to

find the slant height, l.

16. Which has a greater surface area,

the prism or the pyramid? Explain

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. MATHWRITING IN Explain how you can use the height of a pyramid to

find the slant height.

H.O.T. Problems

19. Which is the best estimate for the

surface area of the pyramid?

13.4 ft 13.4 ft

16 ft

A 107 ft 2

B 180 ft 2

C 429 ft 2

D 608 ft 2

20. The net of a paperweight is shown

below. Which is closest to the lateral

surface area of the paperweight?

9.1 cm

7 cm

F 32 cm 2 H 127 cm 2

G 49 cm 2 J 176 cm 2

396 Chapter 7 Measurement: Area and Volume

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