Find the side length of a regular hexagon, given that the area of the hexagon is x^3 square units and the distance from the center of the hexagon to the midpoint of a side is x units
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If you divide a circle (360 degrees) to 6 triangles you will know that each triangle will 'get' 60 degrees around the center. And since every triangle in the hexagon is regular, you will realize those are triangles with equal sides. (equilateral) The distance from the center of the hexagon to the midpoint of a side is essentially the height of one triangle, which also happens to split that triangle into 2 right triangles. So to find the side of the hexagon, we have to calculate tan(30) = h / x so h = x/sqrt(3) and multiply this by 2 to get 2x/sqrt(3). Since the HEXagon has 6 sides, we multiply by a further 6 to get 12x/sqrt(3).
So the first thing to notice is that in a regular hexagon the area can be broken up into three parts, a rectangle in the middle of height 2x and width s (where s is the length of one side), and two trapezoids on the end, of base lengths 2x and s, and height h. The trick is that h is actually equivalent to (2x-s)/2, which becomes apparent if you draw the hexagon out and start labeling everything. From there its just a basic problem to solve, the sum of the area of the parts is equal to the area of the whole: \[2x \times s +2\times 1/2 (2x \times s \times(2x-s)) = x^3\]