The generatrix of the cone, equal to 12 cm, is inclined to the base plane at an angle of 30 degrees. Find the volume of the cone.

Let’s draw the height of the cone. The height of the cone, the generatrix and the radius of the base form a right-angled triangle (the height is perpendicular to the base, and therefore perpendicular to the radius). In a right-angled triangle opposite an angle of 30 ° lies a leg, which is 2 times smaller than the hypotenuse. Therefore, the height of the cone is h = 6 cm.

Find the radius of the base by the Pythagorean theorem:

R = √ (12² – 6²) = √ (144 – 36) = √108.

The volume of the cone is calculated by the formula V = 1/3 * Sbn * h = 1/3 * пR² * h.

V = 1/3 * п * 108 * 6 = 216п.

Answer: the volume of the cone is 216п.