Making a circle into a triangle (Investigation)

You've probably already encountered the formula for the area of a circle: A = \pi r^{2}. However, where did this formula come from? We're going to have a look at how to prove this area formula. 

Archimedes was the first to prove the area formula, but his proof was quite complicated and kind of boring. Instead, we are going to have a look at what is called the "onion proof". Imagine that a circle is an onion, with lots and lots of layers. To find the area of the circle, we could find the area of each of the layers, and add them all up. 

Let's start with a circle with a radius of 10cm, made up of 10 layers each 1 cm thick.

The outermost layer will have a length equal to the circumference

2 \times \pi \times 10 cm and a height of 1 cm.

 

The next layer will be a bit shorter, but will still have the same height. This pattern will continue, until we get to the innermost layer, which will have a height of 1 cm but a very small length. If we stack these layers on top of each other, we get something which looks like a triangle. It's not quite a triangle, because it's got a few pieces missing, but it sure looks like one.

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However, what if we were more accurate, and used a circle with more than 10 layers? Let's say we used 20 layers instead. The edge of the "triangle" would look much less jagged, and the missing pieces (coloured in red in the diagram below) would much smaller.

If we were to do this with an infinite number of tiny layers, then what you would end up with is a perfectly smooth triangle. 

If we have a perfectly smooth triangle, then we can use our triangle area formula to work out the area.

The base of the triangle is equal to the circumference, and the height is equal to 10cm.

Since the area of a triangle is \frac{1}{2} \times base \times height ,the total area is \frac{1}{2} \times 2 \times \pi \times 10 \times 10 , which equals \pi \times 10^{2}.

 

Although we chose this circle to have a radius of 10 cm, we could easily make this into a formula for other circles by just calling the radius r.

In that case, \pi \times 10^{2} becomes a very familiar formula, \pi r^{2}. 

 

This idea of cutting up a complicated shape into infinitely many layers and adding all the layers up is something you'll be seeing again in a few years time, as it forms the basis of integral calculus.