Question

Area of regular hexagon

Answer 1

Check the picture below.

let's recall that a circle has 360°, so that means every two radii coming from the center of the hexagon, will split those 360° in 6 even pieces, 360/6 = 60, namely those two radii make up a 60° central angle, as you see in the picture.

running a perpendicular from the center, we end up with a 30-60-90 triangle, as you see there, and thus we can use the 30-60-90 rule to get the length of "a".

now, keeping in mind that the perimeter of the polygon is simply 4+4+4+4+4+4 = 24.

$\bf \textit{area of a regular polygon}\\\\ A=\cfrac{1}{2}dp~~ \begin{cases} d=apothem\\ p=perimeter\\[-0.5em] \hrulefill\\ d=2\sqrt{3}\\ p=24 \end{cases}\implies A=\cfrac{1}{2}(2\sqrt{3})(24)\implies A=24\sqrt{3}$