Surface Area of a Cone

The total surface area of a cone is the sum of the area of its base and the lateral (side) surface.

The lateral surface area of a cone is the area of the lateral or side surface only.

Since a cone is closely related to a pyramid, the formulas for their surface areas are related.

Remember, the formulas for the lateral surface area of a pyramid is and the total surface area is .

Since the base of a cone is a circle, we substitute 2πr for p and πr2 for B where r is the radius of the base of the cylinder.

So, the formula for the lateral surface area of a right cone is L. S. A. = πrl, where l is the slant height of the cone. 

Example 1:

Find the lateral surface area of a right cone if the radius is 4 cm and the slant height is 5 cm.

L. S. A. = π(4)(5) = 20π ≈ 62.82 cm2

The formula for the total surface area of a right cone is T. S. A. = πrl + πr2.

Example 2:

Find the total surface area of a right cone if the radius is 6 inches and the slant height is 10 inches.

T. S. A. = π(6)(10) + π(6)2 

             = 60π + 36π

             = 96π inches2

             ≈ 301.59 inches2