# 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

$\begin{array}{l}B=\frac{1}{2}\left(5\right)\left(12\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{2}\left(60\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=30{\text{cm}}^{2}\end{array}$

Therefore, the volume is

$\begin{array}{l}V=Bh\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=30\left(10\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=300{\text{cm}}^{3}\end{array}$

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

$\begin{array}{l}\sqrt{{5}^{2}+{12}^{2}}=\sqrt{25+144}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\sqrt{169}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=13\text{cm}\end{array}$

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

$\begin{array}{l}S=2B+Ph\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2\left(30\right)+\left(30\right)\left(10\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=60+300\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=360{\text{cm}}^{2}\end{array}$

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
$L=Ph$
.