The axial sectional area of the cone is 8 and the radius of the base is 2, find the area of the lateral surface of the cone.

The axial section of the cone is an isosceles triangle, the sides of which are equal to the generatrix of the cone, the base is equal to the diameter of the base of the cone.

The axial cross-sectional area is equal to half the product of the height of the cone by the diameter of its base:

S = 0.5 * h * d = 0.5 * h * 2 * r = h * r.

Knowing the values ​​of the axial section area and base radius, we find the height:

h = S / r = 8/2 = 4.

The generatrix of the cone l, the base radius r and the height h form a right-angled triangle, for which, according to the Pythagorean theorem:

l2 = h2 + r2 = 42 + 22 = 16 + 4 = 20;

l = √20 = 2√5.

The area of ​​the lateral surface of the cone is equal to the product of the generatrix and half the circumference of the base:

S = n * r * l = n * 2 * 2√5 = 4п√5 ≈ 28.1.