# Formulas

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation
$2x+1=7$
has the unique solution
$x=3$
because when we substitute 3 for
$x$
in the equation, we obtain the true statement
$2\left(3\right)+1=7$
.

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area
$A$
of a circle in terms of the radius
$r$
of the circle:
$A=\pi {r}^{2}$
. For any value of
$r$
, the area
$A$
can be found by evaluating the expression
$\pi {r}^{2}$
.

### Example 11: Using a Formula

A right circular cylinder with radius
$r$
and height
$h$
has the surface area
$S$
(in square units) given by the formula
$S=2\pi r\left(r+h\right)$
. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of
$\pi$
.

Figure 3. Right circular cylinder

### Solution

Evaluate the expression
$2\pi r\left(r+h\right)$
for
$r=6$
and
$h=9$
.

$\begin{matrix} S\qquad&=2\pi r\left(r+h\right) \\ \qquad& =2\pi\left(6\right)[\left(6\right)+\left(9\right)] \\ \qquad& =2\pi\left(6\right)\left(15\right) \\ \qquad& =180\pi\end{matrix}$
The surface area is
$180\pi$
square inches.

### Try It 11

Figure 4

A photograph with length L and width W is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm2) is found to be
$A=\left(L+16\right)\left(W+16\right)-L\cdot W$
. Find the area of a matte for a photograph with length 32 cm and width 24 cm.

Solution