The area of a circle

We want to find the area of a circle. It can be calculated as equation. Let us explain how we arrived at this formula and the derivation of Pi (equation).

Derivation of Pi

Consider the unit circle which is a circle with radius equation. 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:

circlearea1

By calculating the area of those rectangles, we can approximate the area of the circle. The width equation of the rectangle is decided by us. We only need to calculate its height equation to calculate the area of it as equation. With the radius going from the center to one point on the rectangle, we get a right triangle and can use the Pythagorean theorem (equation) to find equation:

circlearea2

For the first rectangle, we get equation. Solved for equation, we get equation. For the second strip, we get equation and solved for equation, we get equation. The area of the rectangles can then be calculated as:

(1)   equation

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 equation. In our unit circle, equation, so equation.

(2)   equation

Our result of equation 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 equation), 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 equation of rectangles:

(3)   equation

Theoretically, if we use infinitely many rectangles (equation), we can get the exact area of the rectangle.

    equation

The area of the unit circle is called equation. We can approximate equation with a computer to an arbitrary precision by choosing a very large equation. With this the derivation of Pi is complete.

    equation

To find the area of an arbitrary radius circle in terms of equation (equation), we can factor out the equation:

(4)   equation