Surface Area of a Prism

The surface area of the prism can be seen as the sum of two separate parts: the lateral area and the combined area of the two identical bases. The base area, often symbolized as $B,$ can be substituted into the equation. To determine the lateral area of the prism, consider a net of the given prism. Let $a$ be the length of the side of the base, assuming that it is a regular polygon. Notice that the lateral surface consists of rectangles equal to the number of sides in the base. The pentagonal prism shown here has five lateral faces because a pentagon has five sides. The area of each rectangular lateral face is the product of its sides $a$ and $h.$ If there are $n$ rectangular lateral faces in a prism, then the total lateral area is the product of $n$ and the area of one lateral face. Notice that $na$ is the perimeter of the base, which is often denoted by $P.$ Then, the lateral area can be expressed as follows. Therefore, the formula for the surface area is obtained.

Note that although this proof is written for a regular prism, it is also true for a non-regular prism.