Volume and Surface Area of Prisms
Rule

Surface Area of a Prism

Consider a prism with a height base area and base perimeter
Prism

The surface area of the prism is the sum of the two base areas and the lateral area, which can be calculated as the product of the base perimeter and the height of the prism.

Proof

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 can be substituted into the equation.
To determine the lateral area of the prism, consider a net of the given prism. Let be the length of the side of the base, assuming that it is a regular polygon.
Net of the prism
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 and
If there are rectangular lateral faces in a prism, then the total lateral area is the product of and the area of one lateral face.
Notice that is the perimeter of the base, which is often denoted by 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.

Exercises