What is the formula for the surface area of a cone?

Answer 1

Area = πSr + πr²

Where:

S = the slant length of the cone

r = radius of the base

πr² = area of the base of the cone

πSr = area of outside of cone

--------------------------------------------------------------

The surface area of the cone is the sum of the area of the base and the slanted surface.

The area of the base is the area of a circle = πr²

The area of the slanted surface can be calculated by realising that if the slanted surface is cut by a straight line from the apex to the base it can be "unrolled" into a sector of a circle. Thus it is a fraction of the area of a circle. The length of the circumference of the sector is the length of the circumference of the base of the cone; the length of the whole circle from which this is a sector is the circle with radius of the slant height of the cone; thus:

area slanted surface = πS² × ((2πr)/(2πS)) = πSr

Thus the surface area of a cone = area base + area slant surface

→ surface_area_cone = πr² + πSr = πr(r + S)

where r is the radius of the base and S is the slant height.

Answer 2

Area = C(pi)r + (pi)r2

Where:

C = the side length of the cone

r = radius of the base

(pi)r2 = the base of the cone

C(pi)r = outside of cone