The height of the cone is 12; the perimeter of the axial section is 36; calculate the volume of the cone

Let the radius of the circle be equal to X cm, and the generatrix of the cone is equal to Y cm.

Then in a right-angled triangle AOS, by the Pythagorean theorem,

X ^ 2 + OC ^ 2 = Y ^ 2.

X ^ 2 + 144 = Y ^ 2. (1).

By condition, the perimeter of the axial section is 36 cm.

Then 2 * Y + 2 * X = 36.

Y + X = 18. (2).

Let’s solve the system of equations 1 and 2.

Y = 18 – H.

X ^ 2 + 144 = (18 – X) ^ 2.

X ^ 2 + 144 = 324 – 36 * X + X ^ 2.

36 * X = 324 – 144 = 180.

X = R = OA = 5 cm.

Y = AC = BC = 18 – X = 18 – 5 = 15 cm.

Determine the volume of the cone.

V = π * R ^ 2 * OC / 3 = π * 25 * 12/3 = π * 100 cm3.

Answer: The volume of the cone is π * 100 cm3.