The lateral surface area of the cone is 65п, and the radius of the base of the cosine is 5. Find the height of the cosine.

As you know, the area of the lateral surface of the cone is equal to:

S = π * r * l, where

r – base radius,

l – generator of the cone.

Since the area of the lateral surface of this cone is

S = 65 * π, and r = 5, then we can find its generator:

π * 5 * l = 65 * π,

l = 13.

The radius of the base of the cone and its height are the legs of a right-angled triangle, the hypotenuse of which is the generator.

Let the height of the cone be x. Let’s use the Pythagorean theorem:

x² + 5² = 13²,

x² + 25 = 169,

x² = 144,

x = 12.