As most people reading this blog probably know, the area of a circle is given by
where is the ratio of the diameter of a circle to its circumference, and is the radius of the circle. But, how would we go about proving this well known formula? To do so, we can use something called integration, which is a branch of calculus.
The area under a curve
The area under any curve between the x-axis, the curve, and the lines (see the figure below) is given by
where is the function (the curve), and is an infinitesimally small width. Effectively, what we are doing is summing a series of rectangles of area from the lower limit to the upper limit .
The equation of a circle
To do this for a circle, we need to write the equation for a circle. To make things easier we will centre the circle at the origin. In this case, if the circle has a radius the equation which describes this circle is just
Therefore, to find the area of the circle, all we need to do is integrate between , and then multiply the answer by 4 (as we have only found the area of quarter of the circle).
The integration to do this is
This is not an integral which we can do, so we appear to be stuck!
Changing variables to use “polar coordinates”
Luckily for us, there is a way around this problem. Rather than using x-y co-ordinates (more correctly known as Cartesian co-ordinates), we can change the co-ordinate system to something called polar co-ordinates, and when we do this we get an integral that we can do.
To see how to go from Cartesian to polar co-ordinates, consider the figure below.
We write the coordinates in terms of two new variables , where is the radius of the circle and is the angle between the line from the centre of the circle to the point and the x-axis. When we do this, we can write that and .
Then, to determine the area of the quadrant were are positive, we can instead integrate using . To see how we do this, consider the figure below.
One of the reasons the method of using polar coordinates is easier is that we now have only one variable, , as the radius is a constant. When we were using as our variables, moving along the circle involved both variables changing, but with polar coordinates only one variable changes, .
Instead of finding the area of a rectangle and summing those, we instead consider a slice of the circle, where the angle in the slice is , an infinitesimally small angle, and the radius of each slice is . This is shown by the shaded region in the figure above. We then sum these slices between an angle of and . But, when using calculus, we do not use degrees, but rather we have to use radians, which as I explained in this blog, are a more natural unit for measuring an angle.
As becomes infinitesimally small, the slice becomes a triangle, and the area of a triangle is given by . The height is, of course, just the radius , but what about the base? The base is the length of the arc, which you will recall from the definition of a radian is just .
The integral we wish to do is therefore
But, remember, this is the area of just the positive quadrant, so the area of the whole circle is going to be 4 times this, or
just as the famous formula states!
When I derived the acceleration of an object moving in a circle (the centripetal acceleration)
one of the comments stated that I could have derived the same formula using polar coordinates. Now that I have introduced polar coordinates, I will in a future blog re-derive the formula using them.