A regular triangular pyramid is inscribed in a cone, the generatrix of which is equal to 10

A regular triangular pyramid is inscribed in a cone, the generatrix of which is equal to 10 and has an angle of 60 degrees with the base plane. Find the volume of the pyramid.

Consider a right-angled triangle, in which the generatrix of the cone is the hypotenuse, the height of the cone and the radius of the base – the legs. The base radius is opposite an angle of 30 ° and is equal to half the hypotenuse – 5.
By the Pythagorean theorem, we find the height:
H = √ (10² – 5²) = √75 = 5√3.
We find the volume of the cone by the formula:
V cone = H / 3 * π * r² = 5√3 / 3 * 25 * π = 125π / √3.
To find the volume of a regular triangular pyramid inscribed in a cone, we use the ratio of the volumes of these bodies:
V inscribed pyramid / V cone = 3√3 / 4π;
V inscribed pyramid = 125π / √3 * 3√3 / 4π = 93.75.
Answer: The volume of the pyramid is 93.75.