10-5Surface Area of Pyramids and Cones

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Warm UpFind 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

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

Objectives

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.

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.

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

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

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.

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 perpendicularsegment 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

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.

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

= (8)(17)

= 136 in2

Lateral area of a right cone

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.

Find the lateral area and surface area of the right cone.

Use the Pythagorean Theorem to find ℓ.

L = rℓ

= (8)(10)

= 80 cm2

Lateral area of a right cone

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.

ℓ

Example 2C: Finding Lateral Area and Surface Area of Right Cones

The base edge length and slant height of the regular square pyramid are both multiplied by . Describe the effect on the surface area.

Example 3A: Exploring Effects of Changing Dimensions

Example 3A 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 12

= 260 cm2

S = Pℓ + B 12

= 585 cm2