Question

The area of a given hexagon is equal to the area of an equilateral triangle whose perimeter is 36 inches. Find the length of a side of the regular hexagon.

Answer 1

Each side of the triangle is 12 (36/3)
Half of the "bottom", one of the sides, and the height of the triangle form another (right) triangle. The height of the triangle is sqrt(12^2-6^2) = sqrt(108)
The area of the triangle is 6*sqrt(108)/2
The area of the hexagon is the same.
A hexagon can be split into 6 equilateral triangles.
Each triangle has an area of sqrt(108)/2
Now, reversing what I did earlier, we get sqrt(108)/2=y*sqrt((2y)^2-y^2)/2
Solve for y. sqrt(108)=y*sqrt((2y)^2-y^2)
sqrt(108)=y*sqrt(4y^2-y^2)
sqrt(108)=y*sqrt(y^2(4-1))
sqrt(108)=y*y*sqrt(3)
sqrt(108)/sqrt(3)/y^2
y=sqrt(sqrt(108)/sqrt(3))