What is the area of a n sided regular polygon of lenght 'a' on each side?

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  • atsuo
    Lv 6
    1 year ago
    Favourite 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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