Surface Area of

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Surface Area of Pyramids and Cones
10-5
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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Warm Up Find the missing side length of each
right triangle with legs a and b and hypotenuse
c. 1. a 7, b 24 2. c 15, a 9 3. b 40,
c 41 4. a 5, b 5 5. a 4, c 8
c 25
b 12
a 9
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Objectives
Learn and apply the formula for the surface area
of a pyramid. Learn and apply the formula for the
surface area of a cone.
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Vocabulary
vertex of a pyramid regular pyramid slant height
of a regular pyramid altitude of a pyramid vertex
of a cone axis of a cone right cone oblique
cone slant height of a right cone altitude of a
cone
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The vertex of a pyramid is the point opposite the
base of the pyramid. The base of a regular
pyramid is a regular polygon, and the lateral
faces are congruent isosceles triangles. The
slant height of a regular pyramid is the distance
from the vertex to the midpoint of an edge of the
base. The altitude of a pyramid is the
perpendicular segment from the vertex to the
plane of the base.
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The lateral faces of a regular pyramid can be
arranged to cover half of a rectangle with a
height equal to the slant height of the pyramid.
The width of the rectangle is equal to the base
perimeter of the pyramid.
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Example 1A Finding Lateral Area and Surface Area
of Pyramids
Find the lateral area and surface area of a
regular square pyramid with base edge length 14
cm and slant height 25 cm. Round to the nearest
tenth, if necessary.
Lateral area of a regular pyramid
P 4(14) 56 cm
Surface area of a regular pyramid
B 142 196 cm2
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Example 1B Finding Lateral Area and Surface Area
of Pyramids
Find the lateral area and surface area of the
regular pyramid.
Step 1 Find the base perimeter and apothem.
The base perimeter is 6(10) 60 in.
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Example 1B Continued
Find the lateral area and surface area of the
regular pyramid.
Step 2 Find the lateral area.
Lateral area of a regular pyramid
Substitute 60 for P and 16 for l.
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Example 1B Continued
Find the lateral area and surface area of the
regular pyramid.
Step 3 Find the surface area.
Surface area of a regular pyramid
Substitute for B.
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Check It Out! Example 1
Find the lateral area and surface area of a
regular triangular pyramid with base edge length
6 ft and slant height 10 ft.
Step 1 Find the base perimeter and apothem. The
base perimeter is 3(6) 18 ft.
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Check It Out! Example 1 Continued
Find the lateral area and surface area of a
regular triangular pyramid with base edge length
6 ft and slant height 10 ft.
Step 2 Find the lateral area.
Lateral area of a regular pyramid
Substitute 18 for P and 10 for l.
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Check It Out! Example 1 Continued
Find the lateral area and surface area of a
regular triangular pyramid with base edge length
6 ft and slant height 10 ft.
Step 3 Find the surface area.
Surface area of a regular pyramid
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The vertex of a cone is the point opposite the
base. The axis of a cone is the segment with
endpoints at the vertex and the center of the
base. The axis of a right cone is perpendicular
to the base. The axis of an oblique cone is not
perpendicular to the base.
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The slant height of a right cone is the distance
from the vertex of a right cone to a point on the
edge of the base. The altitude of a cone is a
perpendicular segment from the vertex of the cone
to the plane of the base.
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Example 2A Finding Lateral Area and Surface Area
of Right Cones
Find the lateral area and surface area of a right
cone with radius 9 cm and slant height 5 cm.
L ?rl
Lateral area of a cone
?(9)(5) 45? cm2
Substitute 9 for r and 5 for l.
S ?rl ?r2
Surface area of a cone
45? ?(9)2 126? cm2
Substitute 5 for l and 9 for r.
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Example 2B Finding Lateral Area and Surface Area
of Right Cones
Find the lateral area and surface area of the
cone.
Use the Pythagorean Theorem to find l.
L ?rl
Lateral area of a right cone
?(8)(17) 136? in2
Substitute 8 for r and 17 for l.
S ?rl ?r2
Surface area of a cone
136? ?(8)2 200? in2
Substitute 8 for r and 17 for l.
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Check It Out! Example 2
Find the lateral area and surface area of the
right cone.
Use the Pythagorean Theorem to find l.
L ?rl
Lateral area of a right cone
?(8)(10) 80? cm2
Substitute 8 for r and 10 for l.
S ?rl ?r2
Surface area of a cone
80? ?(8)2 144? cm2
Substitute 8 for r and 10 for l.
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Example 3 Exploring Effects of Changing
Dimensions
The base edge length and slant height of the
regular hexagonal pyramid are both divided by 5.
Describe the effect on the surface area.
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Example 3 Continued
base edge length and slant height divided by 5
original dimensions
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Example 3 Continued
base edge length and slant height divided by 5
original dimensions
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Check It Out! Example 3
The base edge length and slant height of the
regular square pyramid are both multiplied by .
Describe the effect on the surface area.
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Check It Out! Example 3 Continued
8 ft
original dimensions
multiplied by two-thirds
585 cm2
260 cm2
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Example 4 Finding Surface Area of Composite
Three-Dimensional Figures
Find the surface area of the composite figure.
Left-hand cone
The lateral area of the cone is L ?rl
?(6)(12) 72? in2.
Right-hand cone
Using the Pythagorean Theorem, l 10 in. The
lateral area of the cone is L ?rl ?(6)(10)
60? in2.
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Example 4 Continued
Find the surface area of the composite figure.
Composite figure
S (left cone lateral area) (right cone
lateral area)
60? in2 72? in2 132? in2
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Check It Out! Example 4
Find the surface area of the composite figure.
Surface Area of Cube without the top side
S 4wh B
S 4(2)(2) (2)(2) 20 yd2
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Check It Out! Example 4 Continued
Surface Area of Pyramid without base
Surface Area of Composite
Surface of Composite SA of Cube SA of Pyramid
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Example 5 Manufacturing Application
If the pattern shown is used to make a paper cup,
what is the diameter of the cup?
The radius of the large circle used to create the
pattern is the slant height of the cone.
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Example 5 Continued
If the pattern shown is used to make a paper cup,
what is the diameter of the cup?
Substitute 4 for l, the slant height of the cone
and the radius of the large circle.
r 2 in.
Solve for r.
The diameter of the cone is 2(2) 4 in.
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Check It Out! Example 5
What if? If the radius of the large circle were
12 in., what would be the radius of the cone?
The radius of the large circle used to create the
pattern is the slant height of the cone.
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Check It Out! Example 5 Continued
What if? If the radius of the large circle were
12 in., what would be the radius of the cone?
Substitute 12 for l, the slant height of the cone
and the radius of the large circle.
r 9 in.
Solve for r.
The radius of the cone is 9 in.
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Lesson Quiz Part I
Find the lateral area and surface area of each
figure. Round to the nearest tenth, if
necessary. 1. a regular square pyramid with base
edge length 9 ft and slant height 12 ft 2. a
regular triangular pyramid with base edge length
12 cm and slant height 10 cm
L 216 ft2 S 297 ft2
L 180 cm2 S 242.4 cm2
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Lesson Quiz Part II
4. A right cone has radius 3 and slant height 5.
The radius and slant height are both multiplied
by . Describe the effect on the surface
area. 5. Find the surface area of the composite
figure. Give your answer in terms of ?.
S 24? ft2