This document provides instruction on calculating the surface areas of pyramids and cones. It begins with key vocabulary terms like vertex, slant height, and altitude. Examples are then given of using the surface area formulas to find the lateral and total surface areas of regular pyramids and right cones when given dimensions like base length and slant height. The effects of changing dimensions on surface area are explored, and a manufacturing application involving cones is presented.

1. Holt Geometry
10-5 Surface Area of Pyramids and Cones10-5
Surface Area of
Pyramids and Cones
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz

2. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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

3. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Learn and apply the formula for the
surface area of a pyramid.
Learn and apply the formula for the
surface area of a cone.
Objectives

4. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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
Vocabulary

5. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.

6. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.

7. Holt Geometry
10-5 Surface Area of Pyramids and Cones

8. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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

9. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Example 1B: Finding Lateral Area and Surface Area
of Pyramids
Step 1 Find the base
perimeter and apothem.
Find the lateral area and surface area of
the regular pyramid.
The base perimeter is
6(10) = 60 in.
The apothem is , so the base area is

10. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Example 1B Continued
Step 2 Find the
lateral area.
Lateral area of a
regular pyramid
Substitute 60 for P and 16 for ℓ.
Find the lateral area and surface area of
the regular pyramid.

11. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Example 1B Continued
Step 3 Find the
surface area.
Surface area of a
regular pyramid
Substitute for B.
Find the lateral area and surface area of
the regular pyramid.

12. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.
The apothem is so the base area is

13. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.

14. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Step 3 Find the surface area.
Surface area of a regular pyramid
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.
Substitute for B.

15. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.

16. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.

17. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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 = πrℓ Lateral area of a cone
= π(9)(5) = 45π cm2
Substitute 9 for r and 5 for ℓ.
S = πrℓ + πr2
Surface area of a cone
= 45π + π(9)2
= 126π cm2
Substitute 5 for ℓ and
9 for r.

18. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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 = πrℓ
= π(8)(17)
= 136π in2
Lateral area of a right cone
Substitute 8 for r
and 17 for ℓ.
S = πrℓ + πr2
Surface area of a cone
= 136π + π(8)2
= 200π in2
Substitute 8 for r
and 17 for ℓ.

19. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Check It Out! Example 2
Find the lateral area and surface area of the
right cone.
Use the Pythagorean Theorem
to find ℓ.
L = πrℓ
= π(8)(10)
= 80π cm2
Lateral area of a right cone
Substitute 8 for r
and 10 for ℓ.
S = πrℓ + πr2
Surface area of a cone
= 80π + π(8)2
= 144π cm2
Substitute 8 for r
and 10 for ℓ.
ℓ

20. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.

21. Holt Geometry
10-5 Surface Area of Pyramids and Cones
3 in.
2 in.
Example 3 Continued
original dimensions: base edge length and
slant height divided by 5:
S = Pℓ + B
1
2 S = Pℓ + B
1
2

22. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Example 3 Continued
original dimensions: base edge length and
slant height divided by 5:
3 in.
2 in.
Notice that . If the
base edge length and slant height are divided by 5,
the surface area is divided by 52
, or 25.

23. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.

24. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Check It Out! Example 3 Continued
original dimensions: multiplied by two-thirds:
By multiplying the dimensions by two-thirds, the
surface area was multiplied by .
8 ft8 ft
10 ft
S = Pℓ + B
1
2
= 260 cm2
S = Pℓ + B
1
2
= 585 cm2

25. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Example 4: Finding Surface Area of Composite Three-
Dimensional Figures
Find the surface area of the
composite figure.
The lateral area of the cone is
L = πrl = π(6)(12) = 72π in2
.
Left-hand cone:
Right-hand cone:
Using the Pythagorean Theorem, l = 10 in.
The lateral area of the cone is
L = πrl = π(6)(10) = 60π in2
.

26. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Example 4 Continued
Composite figure:
S = (left cone lateral area) + (right cone lateral area)
Find the surface area of the
composite figure.
= 60π in2
+ 72π in2
= 132π in2

27. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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

28. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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

29. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.
The area of the pattern is the lateral area of the
cone. The area of the pattern is also of the area
of the large circle, so

30. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Example 5 Continued
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.
If the pattern shown is used to
make a paper cup, what is the
diameter of the cup?

31. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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.
The area of the pattern is the lateral area of the
cone. The area of the pattern is also of the area of
the large circle, so

32. Holt Geometry
10-5 Surface Area of Pyramids and Cones
Check It Out! Example 5 Continued
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.
What if…? If the radius of the large circle were
12 in., what would be the radius of the cone?

33. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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

34. Holt Geometry
10-5 Surface Area of Pyramids and Cones
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 π.
The surface area is multiplied by .
S = 24π ft2
Lesson Quiz: Part II