The generatrix of the cone is 12 cm inclined to the base plane at an angle of 30 °. Find the area of the base of the cone?

The height of the OB of the cone is perpendicular to the plane of its base, and the vertex B is projected to the center of the circle at the base of the cone.

Then the triangle BOS is rectangular, in which OC is the radius of the circle, BC is the generatrix of the cone.

The OB leg is located opposite angle 30, then its length is equal to half the length of the BC hypotenuse.

ОВ = ВС / 2 = 12/2 = 6 cm.

Then, by the Pythagorean theorem, OC ^ 2 = R ^ 2 = BC ^ 2 – OB ^ 2 = 144 – 36 = 108.

OC = R = 6 * √3 cm.

Determine the area of the base of the cone.

Sb = π * R2 = π * 36 * 3 = π * 108 cm2.

Answer: The area of the base of the cone is π * 108 cm2.