Question

A regular pentagon has side lengths of 7 inches. What is the area of the pentagon?

Answer 1

The area of a regular pentagon is equal to one half the apothem times its perimeter, that is,

$A=\frac{1}{2}a\times P$

since each side has length of 7 inches and the pentagon has 5 sides, the perimeter is given by

$\begin{gathered} P=(7\text{inches)}\times5 \\ P=35\text{ in} \end{gathered}$

Then, in order to obtain the apothem, let's draw a picture of the pentagon:

where a denotes the apothem, which is the altitude of the right traingle from below:

So, we can relate the apothem with the side of 3.5 inche and the angle of 36 degrees by means of the tangent function, that is,

$\tan 36=\frac{3.5}{a}$

then,

$a=\frac{3.5}{\tan 36}$

which gives

$\begin{gathered} a=\frac{3.5}{0.7265} \\ a=4.8173\text{ inches} \end{gathered}$

By substituting the perimeter and this result into the area formula, we have

$\begin{gathered} A=\frac{1}{2}a\times P \\ A=\frac{1}{2}4.8173\times35 \end{gathered}$

Then, the answer is:

$A=84.3033in^2$