# The axial section of the cone is a right-angled triangle, the leg of which is 16 cm. Calculate the area of its lateral surface.

Given: cone, where ∠ASВ = 90 °, AS = SB = 16 cm.

It is required to calculate the area of the lateral surface of the cone.

The area of the lateral surface of the cone is equal to the product of the number π by the radius of the circumference of the base and by the length of the generatrix of the cone. The formula for the area (S side) of the lateral surface of the cone: S side = π * R * L, where R is the radius of the base circle, L is the length of the generatrix of the cone.

The length of the generatrix of the cone is known L = AS = SB = 16 cm. It is necessary to determine the length of the radius (R = AO) of the base of the cone.

Since the triangle ASB is an isosceles right-angled triangle, the height SO lowered from the right angle S to the hypotenuse AB will be both the bisector and the median: AO = AB / 2 = R.

According to the Pythagorean theorem, AB2 = AS2 + SB2 = (16 cm) 2 + (16 cm) 2 = 256 cm2 + 256 cm2 = 512 cm2, whence AB = √ (512 cm2) = 16√ (2) cm.means, R = AB / 2 = (16√ (2) cm) / 2 = 8√ (2) cm.

Therefore, S side = π * (8√ (2) cm) * (16 cm) = 128 * π cm2.

Answer: 128 * π cm2.