Calculate the volume of the cone if the generatrix is 30 cm and the height is 24 cm.

A cone is a body that is formed by the rotation of a right-angled triangle around its leg.

The volume of a cone is equal to one third of the product of its base area and height:

V = 1/3 * S * h.

Consider the axial section of this cone. For convenience, we will designate it as ABC. This section has the shape of an isosceles triangle, the sides of which are 30 cm, and the height is 24 cm. This height divides it into two equal right-angled triangles.

Take, for example, the triangle ΔABH. To calculate AH, we apply the Pythagorean theorem:

AB ^ 2 = BH ^ 2 + AH ^ 2;

AH ^ 2 = AB ^ 2 – BH ^ 2;

AH ^ 2 = 30 ^ 2 – 24 ^ 2 = 900 – 576 = 324;

AH = √324 = 18 cm.

Line segment AH is the radius of the base of the cone.

Find the area of ​​the base:

S = πr ^ 2;

S = 3.14 * 182 = 3.14 * 324 = 1017.36 cm2.

Thus:

V = 1/3 * 1017.36 * 24 = 24416.64 / 3 = 8138.88 cm3.

Answer: the volume of the cone is 8138.88 cm3.