Surface Areas of Cones Find the lateral areas of cones. Find surface areas of cones. A cone-shaped teepee in a Shoshone camp, South Pass, Wyoming 1870

LATERAL AREAS OF CONES Cones have the following characteristics: The base is a circle and the vertex is the point V. The axis is the segment whose endpoints are the vertex and the center of the base. The altitude is the segment from the vertex perpendicular to the base of the cone.

The axis is also an altitude LATERAL AREAS OF CONES The axis is also an altitude Slant height Axis Altitude l Oblique Cone Right Cone

LATERAL AREAS OF CONES l l We can use the net for the cone to derive the formula for the lateral area. r l l r

LATERAL AREAS OF CONES l l The lateral region of the cone is a sector of a circle with radius l The arc length of the sector is the same as the circumference of the base, or 2r. The circumference of the circle containing the sector is 2l The area of the sector is proportional to the area of the circle. l r r l

LATERAL AREAS OF CONES l l The area of the sector is proportional to the area of the circle. l area of sector = measure of arc area of circle = circumference of circle area of sector = 2r 2l2 = 2l (l2)(2r) area of sector = 2l area of sector = rl r r l

Key Concept Lateral Area of a Cone If a right circular cone has a lateral of L square units, a slant height of l units, and the radius of the base is r units, then L = rl l r

Example 1 Lateral Area of a Cone LAMPS Diego has a conical lamp shade with an altitude of 6 inches and a diameter of 12 inches. Find the lateral area of the lamp shade. 6 in. 12 in.

SURFACE AREAS OF CONES To find the surface area of a cone, add the area of the base to the lateral area.

Key Concept Surface Area of a Cone If a right circular cone has a surface area of T square units, a slant height of l units, and the radius of the base is r units, then T = rl + r2 l r

Example 2 Surface Area of a Cone 13.6 cm 4.7 cm.