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 $1 : \sqrt{3}: 2$ so the hypotenuse, which is the side of the equilateral triangle, is

$s = \dfrac{2}{\sqrt{3}} (4 \sqrt{3}) = 8$

so an area of a single triangle

$A_\triangle = \frac 1 2 s h = 16 \sqrt{3}$

The entire hexagon is six of these

$A = 6(16 \sqrt{3}) = 96 \sqrt{3}$