This note is a follow up for my circle area proof here
In this note I will prove that the area of any regular polygon can be written as where a is the anothem and p is the perimeter. I will try my best to make this understandable but it is much easier to understand this proof through video.
In the picture above, we see that all regular polygons can be divided into congruent triangles. The area of a triangle is of course bh/2. However, the triangle's height is equal to the apothem of the original polygon. Thus we have a(b/2).
Now, if we look at all the bases of the triangles, they add up to the perimeter of the original polygon. So the total area of the original polygon is the total area of all the triangles which is where n is the number of sides.
Finally, since bn= the perimeter of the polygon, we arrive at the conclusion that is the area of the original polygon.
Thank you for the challenge @JubayerNirjhor: In my next note, I will prove that the area of any regular polygon can be represented as
. Where n is the number of sides and a is the side length. You can find it here