Area of the Circle and Probability

[musicheck]

(Apologies for the lack of latex)
I was thinking today about simple ways of proving that the area of a circle is pi*r^2, and I came up with the following argument using probability. I googled around for similar arguments, and I found nothing. I am curious if there is a deeper and more general result this could be a special case of, or if there if anyone else knows of something relating to this.

We know that the area of a square of side length r is r^2. Consider the unit circle centered at the origin, and a circle of radius R centered at the origin. We randomly select two numbers N and M from [-1,1], with uniform distribution. The probability that N^2+M^2<=1 is thus the ratio of the area of the unit circle to the area of a square of side length 2. Now we randomly select two numbers N' and M' from [-R,R]. The probability that N'^2+M'^2<=R^2 is the ratio of the area of the circle of radius R to the area of a square of side length 2R. However, our random selection from [-R,R] is the same as a random selection from [-1,1] and then multiplying by R. Thus, we can calculate that N'^2+M'^2<=r^2 has the same probability as N^2+M^2<=1. Thus, the area of a circle of radius R is proportional to the area of a square of side length 2R. It follows that the area of a circle of radius R has area pi*R^2, for some constant pi.

 

[mathman]

It is an elementary geometry fact that if you have two figures that are similar, the areas are proportional to the square of the ratio of linear dimension. The more difficult problem is the evaluation of pi, not that it is a constant.

 

[musicheck]

I agree that this is a very simple fact. What struck me was that I had never thought of this method of proof as very relevant to geometry, and I was wondering if people knew whether this method (or a generalization of it) of attacking a problem can yield something more fruitful.

 

[HallsofIvy]

The method you are talking about, more generally called a "MonteCarlo" method is quite well known in mathematics.

 

[musicheck]

Thank you very much HallsofIvy. I had heard that name before, but I didn't know what it was. I'm glad to hear that it is well known, and I look forward to finding a book to read about it.