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

2. 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. Objectives
Learn and apply the formula for the surface area of a pyramid and cone.

4. The vertex of a pyramid is the point opposite the base of the pyramid
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.

5. 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.

6. Lateral Area of a regular pyramid with perimeter, P
and slant height l is _________
L = ½ Pl
Surface Area of a regular pyramid with lateral area L
and base area B is ________________________
S = L + B or S = ½ Pl + B

7. 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

8. The vertex of a cone is the point opposite the base
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.

9. L = rl S = L + B or S = rl + r2 Lateral Area of a right cone
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.
Lateral Area of a right cone
with radius, r and slant
height l is _______
L = rl
Surface Area of a right cone with lateral area L
and base area B is _______________________
S = L + B or S = rl + r2

10. 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. 10

11. 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.

12. Example 2C: Finding Lateral Area and Surface Area of Right Cones
Find the lateral area and surface area of the right cone.
Use the Pythagorean Theorem to find ℓ. 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.

13. Example 3A: Exploring Effects of Changing Dimensions
The base edge length and slant height of the regular square pyramid are both multiplied by . Describe the effect on the surface area.

14. multiplied by two-thirds:
Example 3A Continued
8 ft
10 ft
8 ft
original dimensions:
multiplied by two-thirds:
S = Pℓ + B 12
S = Pℓ + B 12
= 585 cm2
= 260 cm2
By multiplying the dimensions by two-thirds, the surface area was multiplied by .