Question

What is the area of a hexagon with an apothem of 4 square root 3 ?

Answer 1

A hexagon is six equilateral triangles; the apothem is the height of one of them.

The right triangle associated with that apothem has sides in ratio
`1 : √(3): 2` so the hypotenuse, which is the side of the equilateral triangle, is

`s = (2)/(√(3)) (4 √(3)) = 8`

so an area of a single triangle

`A_\triangle = \frac 1 2 s h = 16 √(3)`

The entire hexagon is six of these

`A = 6(16 √(3)) = 96 √(3)`