Name
Class
Date
17.3 Surface Area of Pyramids
and Cones
Essential Question: How is the formula for the lateral area of a regular pyramid similar to the
formula for the lateral area of a right cone?
Resource
Locker
G.11.C Apply the formulas for the total and lateral surface area of three-dimensional figures,
including ... pyramids, cones, ...to solve problems using appropriate units of measure. Also G.10.B
Developing a Surface Area Formula
Explore
The base of a regular pyramid is a regular polygon, and the lateral faces are congruent
isosceles triangles.

The lateral faces of a regular pyramid can be arranged to cover half of a rectangle whose
height is equal to the slant height of the pyramid. Complete the figure by labeling the
missing dimensions.
ℓ
s
s
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s
s
B
Write an expression for the length of the rectangle
in terms of s.

Write an expression for the area of the rectangle
in terms of P and ℓ.

How does the length of the rectangle
compare to P, the perimeter of the
base of the pyramid?

Write a formula for the lateral area L of the
pyramid. (Hint: Use the fact that the lateral
faces of the pyramid cover half of the rectangle.)

Let B be the base area of the pyramid. Write a formula for the surface area S of the
pyramid in terms of B and L. Then write the formula in terms of B, P, and ℓ.
Reflect
1.
Discussion The pyramid in the above figure has a square base. Do your formulas
only hold for square pyramids or do they hold for other pyramids as well? Explain.
Module 17
1037
Lesson 3
Explain 1
Finding the Surface Area of a Pyramid
Lateral Area and Surface Area of a Regular Pyramid
The lateral area of a regular pyramid with perimeter P and slant height
1 Pℓ.
ℓ is L = _
2
The surface area of a regular pyramid with lateral area L and base area
ℓ
_
B is S = L + B, or S = 1 Pℓ + B.
2
Example 1

B
Find the lateral area and surface area of each regular pyramid.
Step 1 Find the lateral area.
Step 2 Find the surface area.
1 Pℓ
Lateral area formula L = _
2
1 (20)(9)
P = 4(5) = 20 in.
=_
2
Multiply.
= 90 in 2
Surface area formula
Substitute the lateral area;
B = 5 2 = 25 in.
S=L+B
9 in.
= 90 + 25
5 in.
= 115 in 2
Add.
Step 1 Find the slant height ℓ. Use the right triangle shown in the figure.
m and
The legs of the right triangle have lengths
Pythagorean Theorem
ℓ2 = a2 + b2
2
Substitute.
=
Simplify.
=
Take the square root of each side.
m.
12 m
ℓ
2
+
10 m
ℓ=
Step 2 Find the lateral area.
Substitute.
1 Pℓ
L=_
2
1
=_
2
=
Multiply.
( )( )
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Lateral area formula
m2
Step 3 Find the surface area.
Surface area formula
S=L+B
2
Substitute.
=
+
Simplify.
=
m2
Reflect
2.
1 Pℓ to find the lateral area of a pyramid whose base is a scalene
Can you use the formula L = _
2
triangle? If so, describe the dimensions that you need to know. If not, explain why not.
Module 17
1038
Lesson 3
Your Turn
Find the lateral area and surface area of each regular pyramid. Round to the nearest
tenth, if necessary.
3.
4.
6 cm
8 ft
ℓ
4 cm
6 ft
Developing Another Surface Area Formula
Explain 2
The axis of a cone is a segment with endpoints at the vertex and the center of the base.
A right cone is a cone whose axis is perpendicular to the base.
Example 2

Justify a formula for the surface area of a cone.
A net for a right cone consists of a circle and a sector of a circle, as shown.
Complete the figure by labeling the missing dimensions.
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Right Cone
ℓ
ℓ
C
D
r
r
B
Consider the shaded sector in the net. Complete the proportion.
Arc length of sector
Area of sector = ___
__
Area of ⊙ C

Multiply both sides of the proportion by the area of ⊙ C. Complete the equation.
Arc length of sector
Area of sector = ___ · Area of ⊙ C
Module 17
axis
1039
Lesson 3

The arc length of the sector is equal to the
circumference of ⊙ D. Therefore, the arc length
of the sector equals 2πr. Complete the equation
by substituting this expression for the arc length
of the sector and by writing the circumference
and area of ⊙ C in terms of ℓ.

L=

Area of sector = _ ·

The area of the sector in Step E is the lateral area
L of the cone. Complete the formula.
Simplify the right side of the equation as much
as possible.
Let B be the base area of the cone. Write a
formula for the surface area S of the cone in
terms of B and L. Then write the formula in
terms of r and ℓ.
Area of sector =
Reflect
In Step D, why is the arc length of the sector equal to the circumference of ⊙ D?
5.
Explain 3
Finding the Surface Area of a Cone
Lateral Area and Surface Area of a Right Cone
The lateral area of a right cone with radius r and slant height ℓ is L = πrℓ.
The surface area of a right cone with lateral area L and base area B is
S = L + B, or S = πrℓ + πr 2.
ℓ
r

A company packages popcorn in paper containers in the
shape of a right cone. Each container also has a plastic
circular lid. Find the amount of paper needed to make
each container. Then find the total amount of paper and
plastic needed for the container. Round to the nearest
tenth.
6 cm
8 cm
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Example 3
ℓ
Step 1 Find the slant height.
ℓ2 = 62 + 82
Pythagorean Theorem
= 100
Simplify.
ℓ = 10
Take the square root of each side.
Step 2 Find the lateral area.
Lateral area formula
Substitute.
Multiply.
Module 17
Step 3 Find the surface area.
L = πrℓ
Surface area formula
= π(6)(10)
= 60π cm
Substitute.
2
Simplify.
1040
S = L + πr 2
= 60π + π(6)
2
= 96π cm 2
Lesson 3
Step 4 Use a calculator and round to the nearest tenth.
The amount of paper needed for the container is the lateral area,
60π ≈ 188.5 c​m ​2​. The amount of paper and plastic needed for the
container is the surface area, 96π ≈ 301.6 c​m ​2​.
B
Step 1 Find the radius.
2
Pythagorean Theorem​
2
 ​ ​ = ​r ​ ​ + ​
2
Simplify.
= ​r ​2​ +
Subtract
= ​r ​2​
from each side.
Step 2 Find the lateral area.
 ​ ​
24 cm
L = πrℓ
( )​​( )​ =
Substitute and simplify. = π ​
Step 3 Find the surface area.
S = L + π​r 2​ ​
=
Substitute and simplify.
25 cm
=r
Take the square root of each side.
r
( )
π c​m ​2​
π + π​
Step 4 Use a calculator and round to the nearest tenth.
The amount of paper needed for the container is the lateral area,
The amount of paper and plastic needed for the container is
2
​  ​ =
​
π c​m ​2​
π≈
π≈
​cm 2​ ​.
​cm 2​ ​.
Reflect
6.
Two right cones have the same radius. A student said that the cone with the greater slant
height must have the greater lateral area. Do you agree? Explain.
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Your Turn
A company makes candles in the shape of a right cone. The lateral surface of each
candle is covered with paper for shipping and each candle also has a plastic circular
base. Find the amount of paper needed to cover the lateral surface of each candle.
Then find the total amount of paper and plastic needed. Round to the nearest tenth.
7.
8.
2 in.
4 in
2.5 in.
6 in
Module 17
1041
Lesson 3
Explain 4
Example 4

Finding the Surface Area of a Composite Figure
Find the surface area of each composite figure. Round to the nearest tenth.
Step 1 Find the lateral area of the cone.
28 cm
The height of the cone is 90 - 45 = 45 cm.
__
90 cm
By the Pythagorean Theorem, ℓ = √28 2 + 45 2 = 53 cm.
45 cm
L = πrℓ
Lateral area formula
Substitute.
= π(28)(53)
Simplify.
= 1484π cm 2
Step 3 Find the area of the base of the cylinder.
Step 2 Find the lateral area of the cylinder.
Lateral area
L = 2πrh
Area of circle
B = πr 2
Substitute.
= 2π(28)(45)
Substitute.
2
= π(28)
Simplify.
= 2520π cm 2
Simplify.
= 784π cm 2
Step 4 Find the surface area of the composite figure.
S = (cone lateral area) + (cylinder lateral area) + (base area)
= 1484π + 2520π + 784π
= 4788π ≈ 15, 041.9 cm 2
B
Step 1 Find the slant height of the pyramid.
By the Pythagorean Theorem, ℓ =
―――――
√
2
2
+
=
yd.
2 yd
Step 2 Find the lateral area of the pyramid.
Substitute.
1
=_
2
Simplify.
=
( )( )
2 yd
2 yd
yd 2
Step 3 Find the lateral area of the rectangular prism.
L = Ph
Lateral area formula
Substitute.
=
Simplify.
=
( )( )
yd 2
Step 4 Find the surface area of the composite figure.
S = (pyramid lateral area) + (prism lateral area) + (base area)
=
=
Module 17
+
+
≈
yd 2
1042
Lesson 3
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1 Pℓ
L=_
2
Lateral area formula
Reflect
9.
How can you check that your answer in Part B is reasonable?
Your Turn
Find the surface area of each composite figure. Round to the nearest tenth.
10.
11.
12 m
8m
4 cm
13 m
6 cm
6 cm
4m
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Elaborate 12. A regular pyramid has a base that is an equilateral triangle with sides 16 inches long. Is it possible to
determine the surface area of the pyramid? If not, what additional information do you need?
13. Explain how to estimate the lateral area of a right cone with radius 5 cm and slant height 6 cm. Is your
estimate an underestimate or overestimate? Explain.
14. Essential Question Check-In How is the formula for the lateral area of a regular pyramid similar to the
formula for the lateral area of a right cone?
Module 17
1043
Lesson 3
Evaluate: Homework and Practice
1.
Multiple Response Which
expression represents the surface
area of the regular square
pyramid shown? Select all
that apply.
t2
ts
A. _ + _
16
2
2.
t2
B. _
16
• Online Homework
• Hints and Help
• Extra Practice
t
s
t2
tℓ
C. _ + tℓ + _
4
2
ℓ
(
t
D. _ _t + ℓ
2 8
)
(
t
E. _ _t + s
2 8
)
Justify Reasoning A frustum of a pyramid is a part of the pyramid with
two parallel bases. The lateral faces of the frustum are trapezoids. Use the
area formula for a trapezoid to derive a formula for the lateral area of a
frustum of a regular square pyramid with base edge lengths b 1 and b 2 and
slant height ℓ. Show all of your steps.
b1
ℓ
b2
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3.
Draw Conclusions Explain why slant height is not defined for an oblique cone.
Find the lateral and surface area for each pyramid with a regular base. Where
necessary, round to the nearest tenth.
4.
12 cm
8 cm
Module 17
6.93 cm
1044
Lesson 3
5.
6.
4 ft
15 ft
6 ft
6 ft
16 ft
16 ft
7.
25 cm
40 cm
Find the lateral and total surface area for each cone. Leave the answer in terms of π.
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8.
9.
22 m
23 cm
14 m
Module 17
23 cm
1045
Lesson 3
10.
11.
24 in.
35 in.
25 in.
24 in.
Find the surface area for the composite shape. Where appropriate, leave in terms of π.
When necessary, round to nearest tenth.
12.
13.
26 m
15 ft
12 m
15 m
8 ft
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32 m
18 ft
Module 17
1046
Lesson 3
14.
15.
17 in.
24 in.
16.
14 cm
15 cm 9 cm
19 cm
7 in.
8 cm
5 cm
17.
10 m
8m
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6m
24 m
18. Anna is making a birthday hat from a pattern that is _
​ 3  ​ of a circle of colored paper.
4
If Anna’s head is 7 inches in diameter, will the hat fit her? Explain.
6 in.
Module 17
1047
Lesson 3
19. It is a tradition in England to celebrate May 1st by hanging cone-shaped baskets of flowers on neighbors’
door handles. Addy is making a basket from a piece of paper that is a semicircle with diameter 12 in. What
is the diameter of the basket?
12 in.
20. Match the figure with the correct surface area. Indicate a match by writing a letter for the correct surface
area in the final column for each shape.
Base
Area
Slant
Height
Regular square pyramid
36 c​m ​2​
5 cm
Regular triangular pyramid
√
​ 3 ​ c​m ​2​
√
​ 3 ​ cm
B. 6.9 c​m ​2​
Right cone
16π c​m ​2​
7 cm
C. 96 c​m ​2​
Right cone
π c​m ​2​
2 cm
D. 138.2 c​m ​2​
Shape
_
_
Surface
Area
A. 9.4 c​m ​2​
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Alamy
21. The Pyramid Arena in Memphis, Tennessee, is a square pyramid with base edge
lengths of 200 yd and a height of 32 stories. Estimate the area of the glass on the
sides of the pyramid. (Hint: 1 story ≈ 10 ft)
Module 17
1048
Lesson 3
22. A juice container is a regular square pyramid with the dimensions shown.
a.
Find the surface area of the container to the nearest tenth.
10 cm
8 cm
b.
The manufacturer decides to make a container in the shape of a right cone that
requires the same amount of material. The base diameter must be 9 cm. Find the
slant height of the container to the nearest tenth.
23. Persevere in Problem Solving A frustum of a cone is a part of the
cone with two parallel bases. The height of the frustum of the cone that is
shown is half the height of the original cone.
a.
Find the surface area of the original cone.
5 cm
20 cm
b.
Find the lateral area of the top of the cone.
c.
Find the area of the top base of the frustum.
10 cm
d.
Use your results from parts a, b, and c to find the
surface area of the frustum of the cone.
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H.O.T. Focus on Higher Order Thinking
24. Communicate Mathematical Ideas Explain how you would find the volume of a
cone, given the radius and the surface area.
25. Draw Conclusions Explain why the slant height of a regular square pyramid must
be greater than half the base edge length.
Module 17
1049
Lesson 3
Lesson Performance Task
The pyramid in the figure is built in two levels.
AC = 200 feet
You have a summer job as an intern archaeologist. The archaeologists you are working
with need to apply a liquid microbial biofilm inhibitor to the pyramid to prevent bacterial
degradation of the stones and have asked you to calculate the volume of inhibitor needed for
the job. You find that you need 36,000 gallons of inhibitor for the top level. How many gallons
will you need for the bottom level? (Keep in mind that you won’t be treating the square bases of
the levels.) Explain how you found the answer.
A
E
B
F
G
C
400 ft
D
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Module 17
1050
Lesson 3