1. 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.

2. 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.

4. 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

5. 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.
The apothem is , so the base area is

6. 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.

7. 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.

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. 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.

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.

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. base edge length and slant height divided by 5: original dimensions:
Example 3 Continued
base edge length and slant height divided by 5:
original dimensions:
Notice that If the base edge length and slant height are divided by 5, the surface area is divided by 52, or 25.

13. 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.

14. Check It Out! Example 3 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 .