Question

Find the area of the regular polygon with the given apothem

Answer 1

`\frac{27\sqrt[]{3}}{2}cm^2`

Step-by-step explanation:

The polygon has 6 sides, therefore, it is a hexagon.

A regular hexagon is made up of 6 congruent equilateral triangles.

• Given that the length of the apothem of the hexagon = 3cm

,

• Therefore, the length of one side = 3cm

First, we find the height of one of the equilateral triangle:

`\begin{gathered} 3^2=1.5^2+h^2 \\ h^2=3^2-1.5^2 \\ h^2=6.75 \\ h=\frac{3\sqrt[]{3}}{2}cm \end{gathered}`

Area of one equilateral triangle

`\begin{gathered} =(1)/(2)*3*\frac{3\sqrt[]{3}}{2} \\ =\frac{9\sqrt[]{3}}{4}cm^2 \end{gathered}`

Therefore, the area of the polygon is:

`\begin{gathered} =6*\frac{9\sqrt[]{3}}{4} \\ =\frac{27\sqrt[]{3}}{2}cm^2 \end{gathered}`