# The height of the cone is 12cm, and its generatrix is 13cm. Find the total surface area of the cone.

A cone is a body created by rotating a right-angled triangle around its leg.

The total surface area of a cone is equal to the sum of the area of its lateral surface and the area of its base:

Sp.p. = πrL + πr ^ 62.

But for this it is necessary to find the radius of the base of the given cone. To do this, consider its axial section, which has the shape of an isosceles triangle, in which the generator is its lateral side, and the diameter is the base.

The height of an isosceles triangle divides it into two equal right-angled triangles, in which the hypotenuse is the generatrix, and the height and radius are the legs. Therefore, to calculate the radius, we apply the Pythagorean theorem:

L ^ 2 = r ^ 2 + h ^ 2;

r ^ 2 = L ^ 2 – h ^ 2;

r ^ 2 = 13 ^ 2 – 12 ^ 2 = 169 – 144 = 25;

r = √25 = 5 cm.

Sp.p. = (3.14 * 5 * 13) + (3.14 * 25) = 204.1 + 78.5 = 282.6 cm2.

Answer: the total surface area of the cone is 282.6 cm2.