# The axial section of the cone is an equilateral triangle. A triangular pyramid is inscribed in the cone, the base of which

**The axial section of the cone is an equilateral triangle. A triangular pyramid is inscribed in the cone, the base of which is a right-angled triangle with legs 12cm and 16cm. Find the height of the pyramid.**

Since a pyramid is inscribed into the cone, at the base of which is a right-angled triangle, the hypotenuse of the triangle at the base of the pyramid is the diameter of the cone, and the lateral edges of the pyramid are the generatrices of the cone.

From the right-angled triangle КМH, according to the Pythagorean theorem, we determine the length of the hypotenuse КМ.

KM ^ 2 = D ^ 2 = KH ^ 2 + MH ^ 2 = 256 + 144 = 200.

KM = 20 cm.

Since, by condition, the axial section of the cone is an equilateral triangle, then ВK = ВМ = КМ = 20 cm.

Then BO is the height of the equilateral triangle and the height of the pyramid. BO = KM * √3 / 2 = 20 * √2 / 2 = 10 * √3 cm.

Answer: The height of the pyramid is 10 * √3 cm.