The height of the cone is 9 cm, and the angle at the apex of the axial section is 60 degrees. Find the volume of the cone.

The volume of the cone is calculated using the following formula:

V = 1 / 3πR²H, where R is the radius of the base of the cone, H is the height of the cone.

Consider a triangle, which is formed by the height, radius and generatrix of the cone. This triangle is rectangular. The generatrix is ​​the hypotenuse in it. The angle between the generatrix and the height is equal to half the angle at the apex of the cone. Let’s find it:

60º / 2 = 30º.

As you know, in a right-angled triangle opposite an angle of 30º lies a leg equal to half of the hypotenuse, that is, the radius is equal to half of the generatrix. We denote the radius as x, then the Pythagorean theorem for a given triangle can be written as follows:

(2x) ² = x² + 9².

Let’s solve this equation:

(2x) ² = x² + 9²,

4x² – x² = 81,

3x² = 81,

x² = 81/3,

x² = 27,

x1,2 = ± √27,

x1,2 = ± 3√3.

In this case, only a positive value is needed, that is, the radius of the base is 3√3 cm.Now we will find the volume of the cone:

V = 1/3 * π * (3√3) ² * 9 = 3 * 27 * π = 81π ≈ 254.47 cm³.

Answer: the volume of the cone is approximately equal to 254.47 cm³.