Methods to derive the formula for Area of Circle

This page describes how to derive the formula for the area of circle by using different methods. The formula can be derived by using a number of methods ౼ some of them require the use of basic calculus while some other are just based on simple mathematics. In this post we are going to discuss some of those which are based on calculus.

Method 1

A circle of radius R can be imagined to be constituted of a large number of thin circular rings/strips (which are concentric) with continuously varying radii as shown below in the image.
The radii of elementary rings vary from 0 to R i.e. 0<r<R. Out of these large number of thin circular stripes consider one such arbitrarily. Assume its radius to be r.
The infinitesimal area of this ring element can be written as dA = (2πr)dr. Now to get the total area of the circle we have to integrate this elementry area using calculus. The limits of integration will be from r=0 to r=R (as r is ranging from 0 to R).

Thus we get the expression of total area of the circle. This was one of the ways to get it. The next method to obtain the same is now discussed below.

Method 2

Instead of imagining the circle as a combination of elemental rings one may also prefer to take the small small sectors as the building blocks of the circle.

Focus on an arbitrarily located sector element which is at an angle ϴ from the reference axis as shown in the adjoining diagram.

As the region occupied by the sector is too small we can approximate it as a triangle with base length Rdϴ and height R (i.e. the radius) and write its small area as
On integrating both sides of the above equation we get the total area:

Method 3

One way of visualising a circle can also be to approx it as being composed of large number of small rectangular two dimensional strips. So let's choose a small rectangular strip inside the periphery of the circle which is at a distance say r from the centre O and subtending a small angle dΦ at the centre O.
The area of the chosen strip is

which again gives the total area. 

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