Given:
Side length = 12 in
To find:
The area of the regular polygon.
Solution:
Number of sides (n) = 6
Let us find the apothem using formula:
[tex]$a=\frac{s}{2 \tan \left(\frac{180^\circ}{n}\right)}[/tex]
where s is side length and n is number of sides.
[tex]$a=\frac{12}{2 \tan \left(\frac{180^\circ}{6}\right)}[/tex]
[tex]$a=\frac{6}{ \tan (30^\circ)}[/tex]
[tex]$a=\frac{6}{ \frac{1}{\sqrt{3} }}[/tex]
[tex]$a=6\sqrt{3}[/tex]
Area of the regular polygon:
[tex]$A=\frac{1}{2}(\text { Perimeter })(\text { apothem })$[/tex]
[tex]$A=\frac{1}{2}(6 \times 12)(6\sqrt{3} )[/tex]
[tex]$A=\frac{1}{2}(72)(6\sqrt{3} )[/tex]
[tex]A=216 \sqrt{3}[/tex]
[tex]A=374.1[/tex] in²
The area of the regular polygon is 374.1 in².
