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