Find the volume of the cone if the radius of its base is 6 dm, and the radius of the sphere inscribed in the cone is 3 dm.

Let’s construct the segment AO1. Let the angle OAO1 = α.

Rectangular triangles OAO1 and ADO1 are equal in leg and hypotenuse, then the angle DAO1 = OAO1 = α, and the angle OAB = 2 * α.

In a right-angled triangle OO1A tgα = OO1 / AO = 3/6 = 1/2.

tg2 * α = 2 * tgα / (1 – tg2α) = 1 / (1 – 1/4) = 4/3.

In a right-angled triangle AOB tg2 * α = ОВ / ОА.

4/3 = OB / 6.

OB = 4 * 6/3 = 8 cm.

Then the volume of the cone is: V = π * AO2 * ОВ / 3 = π * 36 * 8/3 = 96 * π cm3.

Answer: The volume of the cone is 96 * π cm3.