1. Warm Up( Add to HW)
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
c = 25
b = 12

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

3. 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. 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. 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

7. 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

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. Check It Out! Example 2
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.

15. 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.

16. 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:
S = Pℓ + B 12
S = Pℓ + B 12
Notice that The surface area is divided by 52, or 25.

17. 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.

18. 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 .

19. 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 ≈ cm2

20. 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 .
The surface area is multiplied by .
S = 24 ft2