Question

The base of a regular pyramid is a hexagon.

Answer 1

Check the picture below.

since the triangle is a 30-60-90 triangle, we can use the 30-60-90 rule.

now, we know half-side is "b", which is 7, meaning the length of a side in that hexagon is 7+7 or 14.

Since there are 6 sides the "perimeter", namely the sum of its sides is 14*6, or 84.  The length from the center of the regular polygon to a side is the "apothem".

$\bf \textit{area of a regular polygon}\\\\ A=\cfrac{1}{2}ap~~ \begin{cases} a=apothem\\ p=perimeter\\ -------\\ a=7\sqrt{3}\\ p=84 \end{cases}\implies A=\cfrac{1}{2}(7\sqrt{3})(84) \\\\\\ A=294\sqrt{3}$