The Area of a Circle and the Value of Pi

The Area of a Circle and the Value of Pi

Getty Images

The Area of a Circle and the Value of Pi

Getty Images

It is once again Pi Day (March 14 or 3/14 in USA date format). I would just like to point out that the fractional representation of 22/7 is better than the three digits 3.14 thus the date of July 22 could be Pi Day also. If you want many more fun Pi Day posts, here are a few. Yes, Pi is indeed awesome.

The Area of a Circle

So you have a circle. What is the area of that circle? Surely everyone remembers that the area of a circle is:


Where Pi (π) is of course the number and r is the radius of the circle. Where does this formula come from? One method of obtaining this equation is to integrate dxdy over the area of a circle. Well, you probably wouldn't want to do that in cartesian coordinates - but you get the idea.

I recently saw a graphical derivation of the area of a circle. Let's say you start with a circle and break it into 4 wedges. The area of the 4 wedges should be the area of the circle (since that's where they came from).


Maybe you can see where this is going - but what happens if I cut thinner wedges? Here is another way to break it up with even more wedges.


It's really starting to look like a rectangle now. Eventually, it would be almost a perfect rectangle with enough wedges. The vertical side of this rectangle is the radius of the circle and the length of the side is half of the circumference (so, 2πR). Yes, the area of this rectangle would be πR2. That's the area of a circle. Yes, this is sort of cheating. It's cheating because it assumes the circumference is 2πR. But still, it's something.

Another Method to Measure the Area of a Circle

There is a trick to measure the area of a circle. In fact, this was a trick people used in the past to find the area under a curve on a graph (before technology gave us better methods). Suppose I take a piece of paper and I find the mass of the paper. Now I draw a circle and cut the circle out. If I find the mass of the paper cut out circle, the area of the circle will be:


Of course there is a problem here. This goes with the assumption that the area density (mass per unit area) for the paper is fairly constant. If the paper is uneven than this doesn't give a very good value for the area.

Let me get an estimate for the variation in the density of paper. If I start with a stack of printer paper, the sheets all look the same size. I will assume that the uncertainty in the area is very small. Now I can measure the mass of different sheets of paper and get the standard deviation.

Mass of 8 x 11.5 Sheets of Papernewplot-4.png

That's not too bad. The standard deviation of 25 sheets of paper is just 0.5% of the mass of the sheet. Now I will make some circles. If I cut out circles of different diameters and measure the mass of the circle then I can calculate the area. If the area is also supposed to be πR2, I can make a plot of area vs. diameter squared. The slope of this line should be π/4. Here is the plot.

Area of Circles
newplot-5.pngNow relating the slope of this line to π:


Not the best value - but still, it's better than just "3" for a value of π. How could I get a better value? One way would be to use larger circles. If I had even bigger circles, the plot should give a better slope. In fact, the circles I used were quite small (no bigger than a piece of paper). Clearly I couldn't get a circle with a diameter greater than 8 inches (the width of the paper). I suppose I could cut out just half a circle. Or maybe it would be better to use a large sheet of poster board. Maybe you can do that for your next math project.

Maybe next year I will do the same thing with spheres - but I hope not. There is a good chance I will be able to come up with something more interesting before next year's Pi Day.