Question

If the area of a regular nonagon (9-sided polygon) has an area of 783 sq. ft. and anapothem of 14.5 ft., then find the length of each side.

Answer 1

The formula to find the area of a regular nonagon is:

`\begin{gathered} A=(9\cdot s\cdot a)/(2) \\ \text{ Where} \\ A\text{ is the area} \\ s\text{ is the length of anyone side} \\ a\text{ is the length of the apothem} \end{gathered}`

We replace the know values on the above formula and solve for s.

`\begin{gathered} A=783ft^2 \\ a=14.5ft \\ A=(9\cdot s\cdot a)/(2) \\ 783ft^2=(9\cdot s\cdot14.5ft)/(2) \\ 783ft^2=(s\cdot130.5ft)/(2) \\ \text{ Mutiply by 2 from both sides} \\ 783ft^2\cdot2=(s\cdot130.5ft)/(2)\cdot2 \\ 1566ft^2=s\cdot130.5ft \\ \text{ Divide by }130.5ft\text{ from both sides} \\ (1566ft^2)/(130.5ft)=(s\cdot130.5ft)/(130.5ft) \\ 12ft=s \end{gathered}`

Therefore, the length of each side of the regular nonagon is 12 ft.