Surface Areas of Pyramids and Cones

1. S U R FA C E A R E A S O F
P Y R A M I D S A N D C O N E S
S E C T I O N 1 2 - 3

2. E S S E N T I A L Q U E S T I O N S
• How do you find lateral areas and surface areas of
pyramids?
• How do you find lateral areas and surface areas of
cones?

3. V O C A B U L A RY
1. Regular Pyramid:
2. Slant Height:
3. Right Cone:
4. Oblique Cone:

4. V O C A B U L A RY
1. Regular Pyramid:
2. Slant Height:
3. Right Cone:
4. Oblique Cone:
Has a base that is a regular
polygon, and the altitude has an endpoint at
the center of the base

5. V O C A B U L A RY
1. Regular Pyramid:
2. Slant Height:
3. Right Cone:
4. Oblique Cone:
Has a base that is a regular
polygon, and the altitude has an endpoint at
the center of the base
The height of the lateral face

6. V O C A B U L A RY
1. Regular Pyramid:
2. Slant Height:
3. Right Cone:
4. Oblique Cone:
Has a base that is a regular
polygon, and the altitude has an endpoint at
the center of the base
The height of the lateral face
A cone whose axis is also the
altitude (perpendicular to the base)

7. V O C A B U L A RY
1. Regular Pyramid:
2. Slant Height:
3. Right Cone:
4. Oblique Cone:
Has a base that is a regular
polygon, and the altitude has an endpoint at
the center of the base
The height of the lateral face
A cone whose axis is also the
altitude (perpendicular to the base)
A cone whose axis is not an
altitude

8. PA R T S O F A P Y R A M I D

9. PA R T S O F A P Y R A M I D
Vertex

10. PA R T S O F A P Y R A M I D
Vertex

11. PA R T S O F A P Y R A M I D
Vertex
Altitude

12. PA R T S O F A P Y R A M I D
Vertex
Altitude

13. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face

14. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face

15. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face
Lateral Edge

16. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face
Lateral Edge

17. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face
Lateral Edge
Slant Height

18. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face
Lateral Edge
Slant Height

19. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face
Lateral Edge
Slant Height
Base

20. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face
Lateral Edge
Slant Height
Base

21. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face
Lateral Edge
Slant Height
Base
Base Edge

22. PA R T S O F A P Y R A M I D
Vertex
Altitude
Lateral Face
Lateral Edge
Slant Height
Base
Base Edge

23. F O R M U L A S F O R R E G U L A R P Y R A M I D S

24. F O R M U L A S F O R R E G U L A R P Y R A M I D S
Lateral Area of a Regular Pyramid

25. F O R M U L A S F O R R E G U L A R P Y R A M I D S
Lateral Area of a Regular Pyramid
L =
1
2
Pl

26. F O R M U L A S F O R R E G U L A R P Y R A M I D S
Lateral Area of a Regular Pyramid
L =
1
2
Pl
Surface Area of a Regular Pyramid

27. F O R M U L A S F O R R E G U L A R P Y R A M I D S
Lateral Area of a Regular Pyramid
L =
1
2
Pl
Surface Area of a Regular Pyramid
SA =
1
2
Pl +B

28. E X A M P L E 1
Find the lateral area and the surface area of the
square pyramid. The base has a side length of 2.5
cm and a slant height of 5 cm.

29. E X A M P L E 1
Find the lateral area and the surface area of the
square pyramid. The base has a side length of 2.5
cm and a slant height of 5 cm.
L =
1
2
Pl

30. E X A M P L E 1
Find the lateral area and the surface area of the
square pyramid. The base has a side length of 2.5
cm and a slant height of 5 cm.
L =
1
2
Pl
L =
1
2
(2.5)(4)(5)

31. E X A M P L E 1
Find the lateral area and the surface area of the
square pyramid. The base has a side length of 2.5
cm and a slant height of 5 cm.
L =
1
2
Pl
L =
1
2
(2.5)(4)(5)
L = 25 cm2

32. E X A M P L E 1
Find the lateral area and the surface area of the
square pyramid. The base has a side length of 2.5
cm and a slant height of 5 cm.

33. E X A M P L E 1
Find the lateral area and the surface area of the
square pyramid. The base has a side length of 2.5
cm and a slant height of 5 cm.
SA =
1
2
Pl +B

34. E X A M P L E 1
Find the lateral area and the surface area of the
square pyramid. The base has a side length of 2.5
cm and a slant height of 5 cm.
SA =
1
2
Pl +B
SA =
1
2
(2.5)(4)(5)+ 2.52

35. E X A M P L E 1
Find the lateral area and the surface area of the
square pyramid. The base has a side length of 2.5
cm and a slant height of 5 cm.
SA =
1
2
Pl +B
SA =
1
2
(2.5)(4)(5)+ 2.52
SA = 31.25 cm2

36. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.

37. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
a2
+b2
= c2

38. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
a2
+b2
= c2
42
+ 62
= l2

39. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
a2
+b2
= c2
42
+ 62
= l2
16 + 36 = l2

40. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
a2
+b2
= c2
42
+ 62
= l2
16 + 36 = l2
52 = l2

41. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
a2
+b2
= c2
42
+ 62
= l2
16 + 36 = l2
52 = l2
52 = l2

42. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
a2
+b2
= c2
42
+ 62
= l2
16 + 36 = l2
52 = l2
52 = l2
l = 52

43. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.

44. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
SA =
1
2
Pl +B

45. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
SA =
1
2
Pl +B
SA =
1
2
(4)(8)( 52)+ 82

46. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
SA =
1
2
Pl +B
SA =
1
2
(4)(8)( 52)+ 82
SA = 16 52 + 64

47. E X A M P L E 2
Find the surface area of the square pyramid to the
nearest hundredth.
SA =
1
2
Pl +B
SA =
1
2
(4)(8)( 52)+ 82
SA = 16 52 + 64
SA ≈ 179.38 m2

48. PA R T S O F A C O N E

49. PA R T S O F A C O N E
Vertex

50. PA R T S O F A C O N E
Vertex

51. PA R T S O F A C O N E
Vertex
Altitude

52. PA R T S O F A C O N E
Vertex
Altitude

53. PA R T S O F A C O N E
Vertex
Altitude
Slant Height

54. PA R T S O F A C O N E
Vertex
Altitude
Slant Height

55. PA R T S O F A C O N E
Vertex
Altitude
Slant Height
Base

56. PA R T S O F A C O N E
Vertex
Altitude
Slant Height
Base

57. PA R T S O F A C O N E
Vertex
Altitude
Slant Height
Base
Radius

58. PA R T S O F A C O N E
Vertex
Altitude
Slant Height
Base
Radius

59. F O R M U L A S F O R C O N E S

60. F O R M U L A S F O R C O N E S
Lateral Area of a Cone

61. F O R M U L A S F O R C O N E S
Lateral Area of a Cone
L = πrl

62. F O R M U L A S F O R C O N E S
Lateral Area of a Cone
L = πrl
Surface Area of a Cone

63. F O R M U L A S F O R C O N E S
Lateral Area of a Cone
L = πrl
Surface Area of a Cone
SA = πrl +πr2

64. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.

65. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl

66. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1
2
d

67. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1
2
d
r = 1
2
(2.5)

68. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1
2
d
r = 1
2
(2.5)
r = 1.25 in.

69. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1
2
d
r = 1
2
(2.5)
r = 1.25 in.

70. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1
2
d
r = 1
2
(2.5)
r = 1.25 in.
a2
+b2
= c2

71. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1
2
d
r = 1
2
(2.5)
r = 1.25 in.
a2
+b2
= c2
82
+1.252
= l2

72. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1
2
d
r = 1
2
(2.5)
r = 1.25 in.
a2
+b2
= c2
82
+1.252
= l2
64 +1.5625 = l2

73. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1
2
d
r = 1
2
(2.5)
r = 1.25 in.
a2
+b2
= c2
82
+1.252
= l2
64 +1.5625 = l2
65.5625 = l2

74. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1
2
d
r = 1
2
(2.5)
r = 1.25 in.
a2
+b2
= c2
82
+1.252
= l2
64 +1.5625 = l2
65.5625 = l2
l = 65.5625

75. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.

76. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl

77. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1.25 in.

78. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1.25 in. l = 65.5625

79. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1.25 in. l = 65.5625
L = π1.25 65.5625

80. E X A M P L E 3
An ice cream cone has an altitude of 8 inches and a
diameter of 2.5 inches. Find the lateral area of the cone
to the nearest hundredth.
L = πrl
r = 1.25 in. l = 65.5625
L = π1.25 65.5625
L ≈ 31.80 in2

81. E X A M P L E 4
Find the surface area of the cone. Round to the nearest
hundredth.

82. E X A M P L E 4
Find the surface area of the cone. Round to the nearest
hundredth.
SA = πrl +πr2

83. E X A M P L E 4
Find the surface area of the cone. Round to the nearest
hundredth.
SA = πrl +πr2
SA = π(1.4)(3.2)+π(1.4)2

84. E X A M P L E 4
Find the surface area of the cone. Round to the nearest
hundredth.
SA = πrl +πr2
SA = π(1.4)(3.2)+π(1.4)2
SA = 4.48π +1.96π

85. E X A M P L E 4
Find the surface area of the cone. Round to the nearest
hundredth.
SA = πrl +πr2
SA = π(1.4)(3.2)+π(1.4)2
SA = 4.48π +1.96π
SA = 6.44π

86. E X A M P L E 4
Find the surface area of the cone. Round to the nearest
hundredth.
SA = πrl +πr2
SA = π(1.4)(3.2)+π(1.4)2
SA = 4.48π +1.96π
SA = 6.44π
SA ≈ 20.23 in2

87. S O L I D M O D E L
L AT E R A L
A R E A
S U R FA C E A R E A
P R I S M
C Y L I N D E R
P Y R A M I D
C O N E
L = Ph
L = 2πrh
L =
1
2
Pl
L = πrl
SA = Ph + 2B
SA = 2πrh + 2πr2
SA =
1
2
Pl +B
SA = πrl +πr2

88. – R O N A L D R E A G A N
“We can’t help everyone, but everyone can help someone.”
P R O B L E M S E T
p. 843 #1-19 odd, skip #3