A Simple proof of Area of the Circle

Calculating area of geometry with curved lines is more challenging than shapes with straight lines. The area of a circle is typically calculated πr² with Calculus. It's the value we accept and use. But did you know that we can derive the formula A=πr² in a more easier way to see why it is the way it is..

Introduction

By the way, this idea struck to me when I was browsing through my coin collection. Why do they say 'area of a circle is πr²'- I thought. Food for thought anyway :P. I sketched it on a piece of paper and here it is..

A Simple Calculation of Area

Draw a circle assuming a perimeter of 'L', and a radius 'r',

An Analysis of Structure of the Circle

Circle is nothing more than a line drawn at a fixed distance(r) from a fixed point. If we can follow this concept carefully, the area formula can be calculated with some amount of logic.

Because, it is easier for us to deal with straight lines, the area of the circle should be thrown into a manageable shape, a quadrilateral(a shape with 4 sides). Let me explain the logic.


We can represent one side of the quadrilateral to represent 'r'(the radius) because the distance between center and circumference is always a fixed distance in the circle. Lets draw a radius on the circle representing this side.

Now, from the corners of this base just drawn,
Circumference is another side of the quadrilateral. But instead of drawing it curved, we draw it as a straight line representing the same length 'L'


On the inner corner, is the center of the circle. This point  actually doesn't move while drawing a circle. But assuming it does, we can draw the other side of the quadrilateral denoting zero length.(l)




Now we complete the fourth side to get a hypothetical quadrilateral representing the same area of the circle.

Because one of the sides is zero length, this is a triangle. We can deal with the area of a triangle, can't we?

Area _triangle = 1/2 x r x L

Practically, we take the perimeter as '2πr'
Therefore, L = 2πr

Area _triangle = 1/2 x r x 2πr 

Area _triangle = πr²

Because Area _triangle = Area _circle

Area _circle =  πr²

Now that wasn't confusing or was it? Leave me your message in the Comments section. :D


P.S. [2016 January]
To my surprise, this possibility to represent area of a circle by a triangle appears to have been discovered by the great mathematician Archimedes.

Further Reading

There are several other simple ways to get the area formula with similar logic. Here's some links that might take you further up

Area of a disk: Wikipedia

Haha.. Now this one's got some approach :D

Proof of the area of a circle