The area of the lateral surface of the cone is 32π cm2, the radius of its base is 4.

The area of the lateral surface of the cone is 32π cm2, the radius of its base is 4. Find the height and angle between the height and its generatrix.

A cone with a generatrix L and a base radius R has a lateral surface area
S = PiRL.
By condition
Pi * 4L = 32Pi, whence
L = 8 cm.
By the Pythagorean theorem, we find the height of the cone
H = (L ^ 2 – R ^ 2) 0.5 = (8 ^ 2 – 4 ^ 2) ^ 0.5 = 48 ^ 0.5 = 6.93 cm.
The angle between the height and the generatrix is 30 °, since the leg R is equal to half of the hypotenuse L.