What is the surface area of a cylinder based solely on the area of its base?

The surface area of a cylinder is determined by both the lateral area and the area of its bases. To understand why the choice indicating that surface area equals 2πrh + πr² is correct, we should break down the components involved.

First, the base of a cylinder is a circle, and the area of that circle is given by the formula πr², where r is the radius of the base. Since a cylinder has two parallel circular bases, one at the top and one at the bottom, you multiply the base area by 2, resulting in 2πr² for the bases' contribution to the surface area.

The lateral surface area refers to the curved surface that connects the two bases. To visualize or derive this part, if you were to unroll the lateral surface of the cylinder, it would form a rectangle. The height of this rectangle corresponds to the height (h) of the cylinder, and the width corresponds to the circumference of the base, which is given by the formula 2πr.

Thus, the lateral surface area is calculated as the product of the circumference of the base and the height: 2πrh. Therefore, to find the total surface area of the cylinder, you must add