A rectangle with sides of 5 cm and 7 cm rotates around the smaller side. Find the total surface area of a shape of revolution

Rotating the rectangle around its side creates a cylinder. The cylinder surface consists of a lateral surface and two cylinder bases.

In our example, the height of the cylinder is h = 5 cm, the radius of the bases is r = 7 cm.

Let us find the area of the lateral surface of the cylinder:

Side = 2 * Pi * r * h = 2 * Pi * 7 * 5 = 70 * Pi (cm2).

Find the surface areas of the two bases of the cylinder:

Sop = 2 * Pi * r ^ 2 = 2 * Pi * 7 ^ 2 = 98 * Pi (cm2).

Find the total surface area of the cylinder:

S = Sbok + Sbn = 70 * Pi + 98 * Pi = 168 * Pi (cm2).

Answer: the total surface area of the rotation figure is 168 * Pi cm2.