Question

Find the area of the regular polygon with the given radius or apothem.

Answer 1

Answer:

Given:

Apothem = 8√3

Solution:

In the, given attachment you can see that the apothem forms a right angled triangle with one of the sides of the hexagon.

The triangle so formed, is a triangle with 30°, 60°, and 90° which is a property of any right angled triangle in hexagon.

They are in proportion of x - x√3 - 2x, with x√3 being the apothem and x is the short leg.

Therefore,

x√3 = 8√3

=⟩ x = 8√3 / √3

= 8

Therefore, length of the short leg is 8in.

The short leg is half of the side of the hexagon.

Hence, side of the hexagon

= 2x

= 2 × 8

= 16in.

Now, perimeter of the hexagon

= 16 × 6

= 96in.

Now, knowing the perimeter (p) and the apothem (a) of the hexagon, the area of the hexagon can be calculated by the following formula:

Area = 1/2 × p × a

= 1/2 × 96 × 8√3

= 384√3 in²

[Note: You can also calculate the area directly by using the formula (3√3) / 2 × (side)², where you can input the side taken out previously.]

Hope you liked my answer, cheers :)