Given:
[tex]a=4\sqrt{3}[/tex]
To find:
The area of the figure
Solution:
The given figure is regular hexagon.
Number of sides = 6
Using apothem formula:
[tex]$a=\frac{s}{2 \tan \left(\frac{180^\circ}{n}\right)}[/tex]
where a is apothem, s is side length and n is number of sides of the polygon.
[tex]$4\sqrt{3} =\frac{s}{2 \tan \left(\frac{180^\circ}{6}\right)}[/tex]
Multiply by 2 on both sides.
[tex]$8\sqrt{3} =\frac{s}{ \tan {30^\circ}}[/tex]
[tex]$8\sqrt{3} =\frac{s}{ \frac{1}{\sqrt{3} } }[/tex]
[tex]$8\sqrt{3} =\frac{s{\sqrt{3} }}{ 1 }[/tex]
Cancel the common factor [tex]\sqrt{3}[/tex] on both sides, we get
[tex]8=s[/tex]
Side length of the polygon = 8 units
Area of the polygon:
[tex]$A=\frac{1}{2} \times ( \text {apothem }\times \text{ perimeter})[/tex]
[tex]$A=\frac{1}{2} \times ( 4\sqrt{3} \times 6\times 8)[/tex]
[tex]$A=96\sqrt{3}[/tex]
A = 166.27 in²
The area of the figure is 166.27 in².
