A ball is inscribed in a cone with a radius of 9 cm and a generatrix of 15 cm. Find its surface area.

Consider the axial section of this cone. This is an isosceles triangle, the sides of which are 15 cm each, and the base is the diameter of the base of the cone, 18 cm. A circle is inscribed in this triangle, the radius of which is equal to the radius of the ball inscribed in the cone.
Find the radius of the inscribed circle through the semi-perimeter:
p = 1/2 * (a + b + c) = 1/2 * (15 + 15 + 18) = 24 (cm).
r = √ (((p – a) * (p – b) * (p – c)) / p) = √ ((9 * 9 * 6) / 24) = √ (486/24) = √20, 25 = 4.5 (cm).
r = R = 4.5 (cm).
Find the surface area of the ball:
S = 4 * π * R² = 4 * 3.14 * 4.5² = 254.34 (cm²).
Answer: the surface area of the ball is 254.34 cm².