Answer:
471.1 square units
Step-by-step explanation:
A regular nonagon is a 9-sided polygon with sides of equal length.
The apothem of a regular polygon is the distance from the center of the polygon to the midpoint of one of its sides.
Therefore, the given diagram shows a regular nonagon with an apothem of 12 units.
The side length (s) of a regular polygon can be calculated using the apothem formula:
[tex]\boxed{\begin{minipage}{5.5cm}\underline{Apothem of a regular polygon}\\\\$a=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\where:\\\phantom{ww}$\bullet$ $s$ is the side length.\\ \phantom{ww}$\bullet$ $n$ is the number of sides.\\\end{minipage}}[/tex]
Given values:
Substitute the given values into the formula to create an expression for the side length (s):
[tex]12=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{9}\right)}[/tex]
[tex]12=\dfrac{s}{2 \tan\left(20^{\circ}\right)}[/tex]
[tex]s=24 \tan\left(20^{\circ}\right)[/tex]
The standard formula for an area of a regular polygon is:
[tex]\boxed{\begin{minipage}{6cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{n\cdot s\cdot a}{2}$\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the length of one side.\\ \phantom{ww}$\bullet$ $a$ is the apothem.\\\end{minipage}}[/tex]
Substitute the found expression for s together with n = 9 and a = 12 into the formula and solve for A:
[tex]A=\dfrac{9 \cdot 24 \tan(20^{\circ}) \cdot 12}{2}[/tex]
[tex]A=1296\tan(20^{\circ})[/tex]
[tex]A=471.705423...[/tex]
[tex]A=471.7\; \sf square\;units\;(nearest\;tenth)[/tex]
Therefore, the area of a regular nonagon with an apothem of 12 units is 471.1 square units, rounded to the nearest tenth.
