The area of a circle
We want to find the area of a circle. It can be calculated as . Let us explain how we arrived at this formula and the derivation of Pi ().
Derivation of Pi
Consider the unit circle which is a circle with radius . One way of finding its area is to use other geometrical shapes whose area we can already calculate such as a rectangle. Let us decide on a width of the rectangle and place as many as we can inside the circle. The height of the rectangle depends on where it touches the circle. In this example we fit ten rectangles inside the circle:
By calculating the area of those rectangles, we can approximate the area of the circle. The width of the rectangle is decided by us. We only need to calculate its height to calculate the area of it as . With the radius going from the center to one point on the rectangle, we get a right triangle and can use the Pythagorean theorem () to find :
For the first rectangle, we get . Solved for , we get . For the second strip, we get and solved for , we get . The area of the rectangles can then be calculated as:
The same rectangle is present four times in the circle (once in each quarter of it). By adding all areas of the rectangles and multiplying this by four, we can approximate the area of the circle. In our example we fit five rectangles into the circle. Thus, the width is . In our unit circle, , so .
Our result of is fairly imprecise. This is because of all the space in the circle that is not covered by rectangles. We can increase the number of rectangles and this space will become smaller. Hence, the more rectangles we fit into the circle (the smaller the ), the more precise our area approximation will be. By observation of our formula above, we can find a pattern so that we can easily calculate the area for an arbitrary number of rectangles:
Theoretically, if we use infinitely many rectangles (), we can get the exact area of the rectangle.
The area of the unit circle is called . We can approximate with a computer to an arbitrary precision by choosing a very large . With this the derivation of Pi is complete.
To find the area of an arbitrary radius circle in terms of (), we can factor out the :