**What is the average distance between 2 points in a rectangle?**

Say we have a rectangle, and choose any 2 random points within it. We then could calculate the distance between the 2 points. If we do this a large number of times, what would the average distance between the 2 points be?

**Monte Carlo method**

The Monte Carlo method is perfect for this – we can run the following code on Python:

This code will find the average distance between 2 points in a 10 by 10 square. It does this by generating 2 random coordinates, finding the distance between them and then repeating this process 999,999 times. It then works out the average value. If we do this it returns:

This means that on average, the distance between 2 random points in a 10 by 10 square is about 5.21.

**Generalising to rectangles**

I can now see what happens when I fix one side of the rectangle and vary the other side. The code below fixes one side of the rectangle at 1 unit, and then varies the other side in integer increments. For each rectangle it then calculates the average distance.

This then returns the first few values as:

This shows that for a 1 by 1 square the average distance between two points is around 0.52 and for a 1 by 10 rectangle the average distance is around 3.36.

**Plotting some Desmos graphs**

Because I have included the comma in the Python code I can now copy and paste this straight into Desmos. The dotted green points show how the average distance of a 1 by x rectangle changes as x increases. I then ran the same code to work out the average distance of a 10 by x rectangle (red), 20 by x rectangle (black), 30 by x rectangle (purple) and 100 by x rectangle (yellow).

We can see if we continue these points further that they all appear to approach the line y = 1/3 x (dotted green). This is a little surprising – i.e when x gets large, then for any n by x rectangle (with n fixed), an increase in x by one will tend towards an increase in the average distance by 1/3.

**Heavy duty maths!**

There is actually an equation that fits these curves – and will give the average distance, a(X) between any 2 points in a rectangle with sides a and b (a≥b). Here it is:

I added this equation into Desmos, by changing the a to x, and then adding a slider for b. So, when I set b=1 this generated the case when the side of a rectangle is fixed as 1 and the other side is increased:

Plotting these equations on Desmos then gives the following:

Pleasingly we can see that the points created by our Monte Carlo method fit pretty much exactly on the lines generated by these equations. By looking at these lines at a larger scale we can see that they do all indeed appear to be approaching the line y = 1/3 x.

**General equation for a square**

We can now consider the special case when a=b (i.e a square). This gives:

Which we can simplify to give:

We can see therefore that a square of side 1 (a=1) will have an average distance of 0.52 (2dp) and a square of side 10 (a=10) will have an average distance of 5.21 – which both agree with our earlier results.

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