Here is a proof of the area of a circle to satisty the usual questions teachers get all the time when introducing the formula to find the area of a circle: A = Pi × r

Soon or later teachers have to confront kids as they ask, " Where did you get that from ? "

or " Why is the area of a circle Pi times radius squared ? "

We start our proof by having you look at the following figures:

Keep this fact in mind since we will refer back to it later!

Now, consider the following figure. Hang on! we are halfway of completing the proof of the area of a circle :

The area of triangle AOB is 1/2 ( base × height) = 1/2 (s × r)

However, we can make 8 such triangles inside the octagon as show below:

Notice that 8s is equal to the perimeter of the octagon

As stated before, if we increase the number of sides to infinity or a very big number, the resulting n-gon ( The regular polygon which number of sides is a big number) will will almost look like a circle

This means that the perimeter of the octagon will almost be the same as the perimeter of the circle

As a result, the closer the perimeter of the polygon is to the circle, the closer the area of the polygon is to the area of the circle

It is reasonable then to replace 8s by 2 × pi × r, which is the perimeter of the circle, to calculate the area of the polygon or the circle when the number of sides is very big

Doing so we get:

Area of circle or polygon equal = 1/2 r × 2 × pi × r = pi × r

Proof of the area of the circle has come to completion. Any questions? Contact me