 #### How do you find the area of a circle?

The diagram below demonstrates that the area of a circle must be less that 4r². However, it is possible to find the area of a circle by a clever method of dissection. The method is the beginning of the calculus and is attributed to the work of Democritus and Archimedes.

Theorem
The area of a circle, A, is given by A = πr², where r is the radius and π is that perfect proportion.

Proof
Please note that this proof assumes the result, C = π d = π × 2r.

We begin by splitting a circle into sections. Then we drew each piece of the circle side by side, alternating. It can be seen that the new shape is approximately rectangle, but it is perhaps necessary to make more dissections to demonstrate this point. Obviously it is still not rectangular, but it is approaching the perfect rectangle, and so we can argue that the more cuts made, the more rectangular it becomes. It will be noted that the area of the 'rectangle' is half the circumference multiplied by the radius, so the area of the circle, A = ½ C × r = ½ × (π × 2r) × r = πr².

It should be noted that making a finite number of cuts will never form a perfect rectangle, however, if we make infinitely many cuts we face a paradox. If we say that each piece is zero in size, infinitely many pieces of zero will reform to make nothing. On the other hand if we say that each piece is some finite amount, then infinitely many of them will reform to make an infinity of surface; clearly a nonsense. This is the paradox of the infinitesimal; that is, we are dealing with a tiny amount that is neither zero nor is it finite. The key to the paradox seems to lie with how it behaves as we approach infinitely many cuts. We call this method the calculus and it was first used by Democritus around 400 BCE. Archimedes (287-212 BCE) used it to link the circumference and area or a circle, and to find the volume and surface area of a sphere. Many have tried to explain the phenomena, and perhaps two of the most famous mathematicians were the English mathematician, Isaac Newton and the German mathematician, Gottfried Wilhelm Leibniz.