A rectangular triangle with 30 cm and 40 cm legs rotates around the hepotenuse. Find the surface area of the body as it rotates.

The body, which is obtained by rotating a right-angled triangle around its hypotenuse, consists of two cones with one base. In order to find the surface area of ​​such a body, it is necessary to find the area of ​​the lateral surface of each cone and sum these values ​​of the areas. The formula for the lateral surface of the cone is S = πrl, where r is the radius of the base and l is the length of the generatrix. The generatrix of each cone has its own and is equal to one of the legs. The base radius is the height drawn from the vertex of the right angle of our right-angled triangle, and is calculated by the formula h = ab / c, where a, b are the legs, c is the hypotenuse. Let’s find the length of the hypotenuse by the Pythagorean formula and calculate h.

c = √ (a ^ 2 + b ^ 2) = c = √ (30 ^ 2 + 40 ^ 2) = 50 (cm);

h = 30 * 40/50 = 24 (cm).

r = h = 24 cm.

We write down the complete formula for the lateral surface and calculate (π = 3.14):

Spov = S1 + S2 = πr l1 + πrl2 = πr * (l1 + l2) = 3.14 * 24 * (30 + 40) = 3.14 * 24 * 70 = 5275.2 (cm2).