# Prism

A prism is a three-dimensional figure or polyhedron having two faces (called the bases of the prism) which are congruent polygons, and the remaining faces parallelograms .  If the remaining faces are rectangles (as in the figure $2$ nd figure above), then the prism is called a right prism . School level math books usually only talk about right prisms, and sometimes when they say prism they mean right prism.

A triangular prism is a prism with a triangular base, like the figure on the right. A rectangular prism is a prism with a rectangular base. (The cube is a special case.) Similarly, a prism with a $5$ -sided base is a pentagonal prism , a prism with a $6$ -sided base is a hexagonal prism , etc.

### Volume of a Prism

The volume $V$ of a right prism is given by the formula

$V=Bh$ ,

where $B$ is the area of the base and $h$ is the height.

Example:

Find the volume of the prism shown. The height of the prism is $10$ cm. The base is a right triangle with legs of length $5$ cm and $12$ cm, so

Therefore, the volume is

Note that for a rectangular prism, the volume formula becomes simply

$V=lwh$ ,

where $l$ is the length of the base, $w$ is the width of the base, and $h$ is the height of the prism.

### Surface Area of a Prism

The surface area $S$ of a prism is given by the formula

$S=2B+Ph$ ,

where $B$ is the area of the base of the prism, $P$ is the perimeter of the base, and $h$ is the height.

In the example above, by the Pythagorean theorem , the hypotenuse of the triangular base has length

So the perimeter of the base is $5+12+13=30$ cm. We already know the base area is $30$ cm 2 . So,

You may sometimes be called upon to find the lateral surface area of a prism. This just means the surface area of the faces not including the bases, or $L=Ph$ .