Given:
Regular polygon.
Side length = 18 in
To find:
The area of the regular polygon.
Solution:
Let us first find the apothem (length from center to side).
Apothem formula:
[tex]$a=\frac{s}{2 \tan \left(\frac{180^\circ}{n}\right)}[/tex]
where s is side length and n is number of sides.
number of sides (n) = 6
[tex]$a=\frac{18}{2 \tan \left(\frac{180^\circ}{6}\right)}[/tex]
[tex]$a=\frac{9}{\tan (30^\circ)}[/tex]
The value of [tex]\tan \left(30^{\circ}\right)=\frac{\sqrt{3}}{3}[/tex]
[tex]$a=\frac{9}{\frac{\sqrt{3}}{3}}[/tex]
[tex]a=9 \sqrt{3}[/tex] in
Area of regular polygon:
[tex]$A=\frac{1}{2} \times p \times a[/tex]
where p is perimeter of polygon and a is apothem.
[tex]$A=\frac{1}{2} \times (6\times 18) \times 9\sqrt{3}[/tex]
[tex]$A=54 \times 9\sqrt{3}[/tex]
[tex]$A=841.78[/tex] square inches.
The area of the regular polygon is 841.78 square inches.
