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Surface Area of Pyramids and Cones

You can use an envelope to make a triangular pyramid.

Ii/Je,rYOU'l.llEAItN

· To find the lateral area and surface area of a regular pyramid, and · to find the lateral area and surface area of a right circular cone.

Triangular Pyramids

'I M aterla 5:

5 1 1"':713 '8 " x 6'2 " enve Iope ~

li/Ay liS IMPOltfANf

Pyramids and cones have been used in structures since ancient times. · Seal a 3%" X

~

6~"

envelope. Then draw both diagonals of the envelope.

straightedge

<\

scissors

· Fold and crease along each diagonal. Then fold and crease along the perpendicular bisector of the long edge of the envelope. Label the point of intersection of the diagonals as point A.

..

· Carefully cut from the top of the envelope along each diagonal to point A. Remove the triangular piece.

'

· Open the envelope. Fold along the perpendicular bisector so that the top corners come together. Tuck one corner inside the other and push until the corner meets the bottom edge. The solid formed is a triangular pyramid.

Your Turn

a. What shape is each face of the pyramid? b. Is each face of the pyramid congruent? Explain. c. Suppose that the face with the letter A on it is the base of the pyramid. What do you observe about the other faces? d. Repeat the activity using a

4i"

X

9~"

envelope and describe your results.

Lesson 11-4

Surface Area of Pyramids and Cones

599

A triangular pyramid is just one type of pyramid. A pyramid has the following characteristics. · All the faces except one intersect at a point called the vertex. · The base is the face that does not intersect the other faces at the vertex. The base is always a polygon. · The faces that intersect at the vertex are called lateral faces and form triangles. The edges of the lateral faces that have the vertex as an endpoint are called lateral edges. · The altitude is the segment from the vertex perpendicular to the base.

If the base of a pyramid is a regular polygon and the segment whose endpoints are the center of the base and the vertex is perpendicular to the base, then the pyramid is called a regular pyramid. In a regular pyramid, the segment whose endpoints are the center of the base and the vertex is also the altitude.

rlateral edge lateral face

slant

height

regular pyramid

In a regular pyramid, all of the lateral faces are congruent isosceles triangles. The height of each lateral face is called the slant height e of the pyramid. The figure below is a regular hexagonal pyramid. Its lateral area L can be found by adding the areas of all its congruent triangular faces as shown in its net.

L

= = =

+se ++se + +se ++se + +se + +se 1 "2(s + S + S + S + S + s)e

+Pe

p = (.

s

+

S -r- ' -

s)

This suggests the following formula.

Lateral Area of a Regular Pyramid 600

Chapter 11

If a regular pyramid has a lateral area of L square units, a slant height of f units, and its base has a perimeter of P units, then L = ~pe.

Investigating Surface Area and Volume

Example

o

The Luxor Hotel in Las Vegas, Nevada, is shaped like a gigantic black glass pyramid. The base of the pyramid is a square with edges 646 feet long. The hotel is 350 feet tall. Find the area of the glass on the Luxor.

The area of the glass is the lateral area of the pyramid.

Architecture

eft

The altitude and the slant height are a leg and the 646 It hypotenuse of a right triangle. The other leg is half of the measure of a side of the base. Use the Pythagorean Theorem, c 2 = a 2 + b 2, to find the slant height of the pyramid.

e2 = 350 2 + (t· 646)2 e2 = 226,829 e = V 226,829 or about 476.3 ft

L =

.l.pe 2

= 4 . 646)(476.3) = 615,379.6

t(

. i i....~:

i

The area of the glass on the Luxor is about 615,380 ft 2.

.

The surface area of a regular pyramid is the sum of its lateral area and the area of its base.

Surface Area of a

Regular Pyramid

If a regular pyramid has a surface area of T square units, a slant height of e units, and its base has a perimeter of P units and an area of B square units, then T= ~pe + B.

Example

A regular pyramid has a slant height of 13 centimeters and a height of 12 centimeters. If the base is a regular pentagon, find the surface area of the pyramid.

The slant height,

the altitude, and the apothem form a right

triangle. Use the

Pythagorean Theorem

to find the length of

the apothem.

13 2 25

=

13cm

12cm

12 2 + a 2

=

a2

(col/tlnued on the next page)

Lesson 11-4 Surface Area of Pyramids and Cones

5=a

601

You can review the trigonometriC functions in Lesson 8-3.

Now find the length of the sides of the base of

the pyramid. The central angle of the pentagon

360 measures -5- or 72°. Let a represent the measure of the angle formed by a radius and

72

the apothem. Thus, a = "2 or 36.

0

Use trigonometry to find the length of the sides.

-is

tan 36°

= =

L

5

5 tan 36°

10 tan 36°

15 2

= 5

~ 5

7.3

Now use the formula to find the surface area of the pyramid.

T=~Pe+B

~

~

2 (5· 7.3)(13) + 2(5 . 7.3)(5) 328.5

I I } B = :;Pa

:

The surface area of the pyramid is about 328.5 cm 2.

.

The figure at the right is a circular cone. Its base is a circle, and its vertex is at V. Its axis, VX, is the segment whose endpoints are the vertex and the center of the base. The segment that has the vertex as one endpoint and is perpendicular to the base is called the altitude of the cone. A cone whose axis is also an altitude is caJled a right cone. Otherwise, it is called an oblique cone. The cone above at the right is an oblique cone, and the cone at the right is a right cone. The measure of any segment joining the vertex of a right cone to the edge of the circular base is called its slant height e. The measure of the altitude is the height h of the cone. You will use formulas similar to the formulas for finding the lateral area and the surface area of a regular pyramid to find those same measures for a right cone. In the net for a cone, the region of the cone that is not the base is a sector of a circle whose radius is the slant height e.

oblique cone

r

right cone

The area of the sector is proportional to the area of the circle. Notice that the arc length of this sector is equal to the circumference of the base of the original cone, 2'TTr.

area of sector area oi circle area of sector 7ff2 measure of arc

circumference of circle

27fr

27fe

=

'TT

area of sector

602

Chapter 11 Invesiigafing Surface Area and Volume

re

Multiply each side by rrf 2.

Lateral Area and Surface Area of a Right Circular Cone

If a right circular cone has a lateral area of L square units, a surface area of T square units, a slant height of f units, and the radius of the 'base is r units, then L = 7l'rf and T = 7l'rf + 7l'r2 ·

Example

Find the lateral area and the surface area of a cone if the slant height is 13 feet and the diameter of the base is 10 feet.

If the diameter is 10 feet,

10 It

13 It

then the radius is

"2 x 10 or 5 feet.

L

=

1

7I"re 6571"

T

=

7I"re + 7I"r2 9071"

.

= 71"(5)(13)

=

= 71"(5)(13) + 7I"(5i

=

= 204.2 ft2

= 282.7 ft 2

:

. ;HECK FOR UNDERSTANDING ~

Communicating Mathematics

Study the lesson. Then complete the following.

1. Compare and contrast the lateral edges of a pyramid and those of a prism.

2. Analyze the change in the shape of the base of a regular pyramid as the number of sides increases.

a. What shape is it? b. As this transformation takes place, what happens to the shape of the pyramid?

3. Sketch a regular pyramid. Which is longer, a lateral edge or i:Pe slant height? Explain. 4. Describe a cone in which the axis is not also the altitude of the cone.

5. Cut the side of a cone-shaped drinking cup from the brim to the vertex. Flatten out the cup. a. What is the shape of the surface? b. Measure the radius and the angle formed by the cut edges.

c. Find the area of the surface. How does the area relate to the formula for

the lateral area of a cone?

Guided Practice

Determine whether the condition given is characteristic of a pyramid or prism, both, or neither.

6. The lateral faces are parallelograms. 7. It can have as few as five faces.

Lesson 71-4 Surface Area of Pyramids and Cones

603

Find the lateral area and the surface area of each regular pyramid or right cone. Round to the nearest tenth.

8.

9.

10.

8ft

3m

11. Find the surface area of the solid at the right.

Round to the nearest tenth.

t4

--- ___ (6 in. ---.... ----

.....

in.

16 '0

12. The base of a rectangular pyramid is 15 inches long and 8 inches wide. The height is 4 inches. Find the surface area of the pyramid if alL of the lateral edges are congruent. 13. History Historians believe that the Great Pyramids of Egypt were once covered with gold or white stones that have worn away or have been removed to be used for other purposes. The diagram at the right shows the approximate dimensions of the Great Pyramid of Khufu. Find the surface area

that would have to have been covered.

756 ft 756 ft.

'~XERCJSES

Practice

Determine whether the condition given is characteristic of a pyramid or prism, both, or neither.

14. There is exactly one base. 15. It always has an even number of faces. 16. There are two bases. 17. The lateral faces are triangles. 18. It has the same number of lateral faces as vertices. 19. There can be as few as four faces. 20. The number of edges is always even.

Find the lateral area and the surface area of each regular pyramid or right cone. Round to the nearest tenth.

21.

(---"-:':":':"'--.

I

8 in.

22.

I

I

I I I

18 in.

I I

I

,////

I

I

..... L

I

_

"

604

Chapter 11 Investigating Surface Area and Volume

10 em

24.

12 em

12em

Find the surface area of each solid. Round to the nearest tenth.

27.

5ft

28.

30. A regular pyramid has a slant height of 13 feet. The area of its square base

is 100 square feet. Find its surface area.

31. fn the given cube, A, B, and C

are the vertices of the base of the pyramid with vertex D. If the edge of the cube is 8 units long, find the lateral area and the surface area of the pyramid.

32. A frustum is the part of a solid that remains after the top portion of the

solid has been cut off by a plane parallel to the base. c. The figure below is a frustum b. Find the surface area of the of a regular pyramid. Find its frustum of a cone shown below. lateral area.

Critical hinking

33. If you were to move the vertex of a right cone down the axis toward the

center of the base, what would happen to the lateral area of the cone? Be as specific as possible and demonstrate the validity of your answer with a series of diagrams.

34. Dwellings The largest tepee in the United States belongs to Dr. Michael Doss of Washington, D.C. Dr. Doss is a member of Montana's Crow Tribe. The tepee is a right cone with a diameter of 42 feet and a slant height of about 47.9 feet. How much canvas was used to cover the tepee?

35. Art In 1921, Italian immigrant Simon Rodia bought a home in Los Angeles, California, and began building conical towers in his backyard. The structures are made of steel mesh and cement mortar with no rivets, bolts, or welds. The first tower completed was the East Tower, which stands 55 feet high. The diameter of the base of the East Tower is 8+ feet. Find the lateral area of the tower.

Lesson 11-4 Surface Area of Pyramids and Cones

Applications and Problem Solving r--_ _

60S

36. History The Cahokia Mounds stand close to East St. Louis, Illinois, where there was once the largest ancient city in America, north of Mexico. The inhabitants began building 120 earthen pyramids there around A.D. 600. The largest mound, Monk's Mound, has a height of 30.5 meters and a rectangular base with sides 216.6 and 329.4 meters long. Find the lateral area of Monk's Mound. (Hint: Is Monk's Mound a regular pyramid?)

Mixed Review

37. If the lateral area of a right rectangular prism is 784 square centimeters, its length is three times its width, and its height is twice its width, find its surface area. (Lesson 11-3) 38. From the views given below, draw a corner view. (Lesson 11-1)

top view

left view

front view

Eb

right view

39. Find the area of a regular nonagon with an apothem 12.2 centimeters long and a side 10.4 centimeters long. (Lesson 10-5) 40. The diameter of a circle measures 10 centimeters, and the length of a chord is 8 centimeters. Find the distance from the chord to the center of the circle. (Lesson 9-3J 41. Photography Kandhi is taking pictures at the zoo. If a gnu is 4.3 feet tall, the film is 1 inch from the camera lens, and the camera lens is standing 5 feet from the gnu, how tall will the gnu's image be on the film? (Lesson 7-5) 42. Graph ( - 6, - 3) and (1, 8). Then draw the line that passes through the points. Through which quadrants does the line pass? (Lesson 1-1)

i

.'

Algebra

43. Translate the sentence The quantity x is equal to 18 more than the square of b into an equation. 44. Use elimination to solve the system of equations.

2x + 4y

= =

-14 23

3x - 5y

1. The corner view of a figure is given at the right. Draw the top, left, front, right, and back views of the figure, (lesson 11-1) 2. Use isometric dot paper to draw a corner view of a rectangular solid 2 units high, 4 units long, and 2 units wide. Then draw a net of the solid on rectangular dot paper. (Lesson 11-2]

(OJ]

:~

Find the lateral area and the surface area of each solid below. Round to the nearest tenth. (lesson 11-3)

3.

12 in.

/

/

4.

6 in.

/

,,

6

.

'C

--

__ '-___

---.

3.1 em

5. Architecture The Transamerica Tower in San Francisco is a regular pyramid with a square base that is 149 feet on each side and a height of 853 feet. Find its lateral area. [Lesson 11-4)

606

Chapter 11

Investigating Surface Area and Volume

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