What is the formula for the surface area of a closed cylinder?

The formula for the surface area of a closed cylinder takes into account both the curved surface area and the area of the two circular bases at the top and bottom.

To derive this, the surface area can be split into two parts:

1. The curved surface area, which wraps around the sides of the cylinder. This area is calculated using the formula (2\pi rh), where (r) is the radius and (h) is the height of the cylinder.

2. The area of the two circular bases. Each base has an area of (πr²), so for two bases, you multiply this area by 2, resulting in (2πr²).

Combining these two components gives the complete formula for the surface area of a closed cylinder:

[

A = 2πr² + 2πrh

]

This formula accounts for both the top and bottom circular surfaces in addition to the side surface. Thus, the correct answer accurately represents the full surface area of the cylinder.