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Regular figures are gloriously simple, once you get the idea.
1) From the center of the figure, draw ALL the apothems and the segments to ALL the vertices.
2) Notice that ALL the apothems form right angles with the sides they intersect.
3) If you look at the TWO segments to adjacent vertices, you should see a lovely isosceles triangle with the included side of the regular figure as its base.
4) Looking at this isosceles triangle, you should see that the included apothem exactly bisects the triangle - AND the included side of the regular figure.
5) If you look at ALL pairs of consecutive segments to vertices, you will see that this pattern of bisected isosceles triangles repeats over and over and over.
6) All you have left to do is to calculate the area of one of the right triangles. The height is the apothem and the base is (1/2) the side of the regular figure. Extrapolate this result to the entire figure.
7) You never will have to draw all the apothems or segments to vertices again. Now you have seen it.