Question

What is the area of the regular polygon?

Answer 1

Answer:

$\frac{27\sqrt{3} }{4} m^2$

Step-by-step explanation:

The area of a regular polygon is given by

$Area=\frac{1}{2}ap$

where $a$ is the apothem and $p$ is the perimeter of the triangle.

The interior angle of a regular hexagon is

$180-\frac{360}{6} =120\degree$

The line from the center to the given vertex bisects the 120 degree angle.

The hexagon is therefore divided into 6 equilateral triangles.

The apothem can be found using $\sin(60\degree)=\frac{a}{3}$

This implies that;

$a=3\sin(60\degree)$

$a=\frac{3\sqrt{3} }{2}$

Recall that an equilateral triangle has all sides equal hence the perimeter of the regular hexagon is $6\times 3=18m^2$

The area of the regular hexagon now becomes;

$Area=\frac{1}{2}\times \frac{3\sqrt{3} }{2}\times 18m^2$

This simplifies to;

$Area= \frac{27\sqrt{3} }{2}m^2$