What formula is used to calculate the surface area of a cylinder?

The formula for calculating the surface area of a cylinder incorporates both the lateral surface area and the areas of the circular bases. It is derived from the geometric properties of a cylinder, which consists of two circular bases and a curved surface.

The correct expression consists of two parts: the lateral surface area and the area of the two bases. The lateral surface area is represented as the circumference of the base multiplied by the height, which is (2\pi r \times h). Additionally, each base contributes an area of (\pi r^2), and since there are two bases, this is represented as (2\pi r^2). These components combined lead to the formula:

[ \text{Surface Area} = \text{Lateral Area} + 2 \times \text{Area of Base} = 2\pi rh + 2\pi r^2. ]

It can also be simplified to show the total as (2\pi r(h + r)), which maintains the necessary relationships between radius, height, and surface area.

This understanding clarifies why the other formulas don’t represent the surface area correctly, either by missing components or misrepresenting the relationships between height and radius in cylindrical geometry