The height of the cone is 6 cm, the angle at the apex of the axial section is 90, find the area of the side surface of the cone.

Since the angle at the apex of the axial section is 90, the axial section of the cone is a right-angled, isosceles triangle ABC.

The height of the OC of the cone is the height, the bisector and the median of the triangle ABC, then in the right-angled triangle AOC the angle is ACO = 90/2 = 450, and therefore the triangle AOC is right-angled and isosceles, AO = OC = 6 cm.

Let us determine the length of the generator of the AC. AC ^ 2 = AO ^ 2 + OC ^ 2 = 36 + 36 = 72.

AC = √72 = 6 * √2 cm.

Then S side = n * AO * AC = n * 6 * 6 * √2 = n * 36 * √2 cm2.

Answer: The lateral surface area is equal to n * 36 * √2 cm2.