+411 A regular hexagon has a perimeter of $p$ (in length units) and an area of $A$ (in square units). If $A = 3,$ then find the side length of the hexagon.
+34 We can first find the area of a hexagon. If we break it into 6 equilateral triangles, we can calculate their areas. Let's say the side length of the regular hexagon is \(s.\) The area of one of the equilateral triangle is \(\frac{s^2 \sqrt{3}}{4}\), so the area of the entife hexagon is \(\frac{3s^2 \sqrt{3}}{2}.\) Because we know the area is 3, we can solve for s:
\(\frac{3s^2 \sqrt{3}}{2} = 3 \) ==> \(3 s^2 \sqrt{3} = 6\) ==> \(s^2 \sqrt{3} = 2\) ==> \(3 s^2 = 2 \sqrt{3} \) ==> \(s^2 = \frac{2 \sqrt{3}}{3}\)
This is best I can go. It get's really complicated if you go even further. Please let me know if I made a mistake.
