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- atsuoLv 61 year agoFavourite answer
Think regular n-polygon , the length of its one side is a .

If we connect the center of n-polygon and all vertices then

the n-polygon is divided into same n isosceles triangles .

The base of one isosceles triangle is a , and let the height of it be h .

The top angle is 2π/n , so we can find

tan(π/n) = (a/2)/h

So

h = (a/2)/tan(π/n)

The area of the isosceles triangle is

(1/2)ah = (a^2)/(4tan(π/n))

So the area of the n-polygon is

n*(a^2)/(4tan(π/n))

For example , let n = 4 . tan(π/4) = 1 so the area of regular 4-polygon (a square) is

4*(a^2)/4 = a^2

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