Can some one tell me if this formula for any regular polygons is correct?

Question

n= number of sides
s=side length

(1/2*s/tan(180/n)*1/2*s*n)

i found it but i don't know if its right

Answer 1

smartprimate's advice is very good. More specifically, put a point in the center of the regular n-gon. Then the area is the sum of the areas of the n triangles formed by connecting the point in the center to all the vertices of the n-gon. The area of the polygon then becomes:

n*area of triangle =
n*(1/2)*base*height

Let the base just be the same of the n-gon, s. Then to find the height use a little trigonometry. Specifically, look at the right triangle formed the center point, a vertex of the n-gon and the midpoint of one of its adjacent sides. The angle corresponding to the center vertex is 180/n, and the length of the opposite side is just s/2, so the length of the line from the midpoint of a side to the center is 1/2*s/tan(180/n). So this is the height of the n triangles.

So n*(1/2)*base*height =
n*(1/2)*s*(1/2)*s/tan(180/n)

which is the formula you found!

Answer 2

Try it for easy known polygons - equilateral triangle, square. If it works for those, try it for less symmetric polygons.
To work out the formula for yourself, divide a polygon into triangles and use that to construct a formula.

Answer 3

i used autocad to draw polygons and finding its area and compared it with formula its right + or - 0.5 accu.