The cylinder and the cone share a common base and height. What is the generatrix of the cone

The cylinder and the cone share a common base and height. What is the generatrix of the cone if the generatrix of the cylinder is 12 and the diameter of the base is 10?

A cylinder is formed by rotating a rectangle around a side, and a cone by rotating a right-angled triangle.

Consider the axial section of these figures, for convenience we denote them:

ABCD – axial section of the cylinder;

AOD is the axial section of the cone.

The length of the generatrix of the cylinder is equal to the length of its height:

h = l = 12 cm.Since the height of the cylinder and the cone is common, and the radius is common, then using the Pythagorean theorem, one can find the length of the generatrix of the cone, since the triangle AOK is rectangular:

AO ^ 2 = OK ^ 2 + AK ^ 2;

AO ^ 2 = 10 ^ 2 + 12 ^ 2 = 100 + 144 = 244;

AO = √244 = 15.6 cm.

Answer: the generatrix of the cone is 15.6 cm.