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# Area of circle

Ever wondered how can we prove that area of a circle is πr² ? So many times we have used this formula but very few books included the fundamental proof of the area of circle.  Here in this blog we try to prove the same using the concepts of integral calculus: Consider the following diagram i.e an circle inscribed within an octagon :

polygon in circle

polygon in circle

We know that the area of triangle is 1/2* base * height. Also we know that circumference of circle is 2πr. If we see the above diagram it’s clear that as we gradually increase the number of sides of polygon to say n-gon, it will be very closely matching the circle and thus, in that case the perimeter of circle will be equal to the perimeter of the n-gon. Let dr denote the length of side of the n-gon. This polygon will have n triangles and thus we have 1/2 * n *  r * dr = area of one of the triangles( as seen above OAB) of the polygon. As n increases n * dr will move close to the perimeter of circle i.e 2πr Thus we get 1/2  * n  *  r  * dr = 1/2 * r * 2πr =  πr² and hence we have the proof