Written by Ruqiyah Patel

Down the street from the Australian Academy of Science is a beloved donut shop.

While walking to the donut shop the reSolve team wondered: which of these two routes is shorter: red or green?

Just by looking at the map it seem reasonable to estimate that the distance in red (the *outer distance*) is shorter than the distance in green (the *inner distance*). What if the donut shop was located somewhere else on the same circuit? If the donut shop was next door, the outer distance would definitely be shorter.

But if the donut shop was on the far side of the circuit, it looks like the inner distance could be shorter.

The reSolve team decided that they wanted to find an answer to this question: *how far away* does the donut shop have to be before the *inner distance* is shorter than the *outer distance*?

We simplified the question to be about travelling around two concentric circles. When is it faster to travel around the outer circle, and when is it faster to take the path along the inner circle? (It is just a coincidence that this model resembles a donut!)

At first we thought the answer might change based on the radius of the inner circle. If the inner circle was very large, then the inner distance might be similar to the outer distance (see the diagram on the left). If the inner circle was smaller, than the inner distance might be significantly different from the inner distance (see the diagram on the right).

To simplify the problem, we set a radius for the inner circle: ^{1}⁄_{2} the radius of the outer circle. Then we tried to solve the question algebraically.

We experimented with a few different values for the radius of the outer circle and started to find some results that surprised us – a pattern we were not expecting!

It seemed like the answer to our question might be more general that we were expecting. It might not depend on the radius of the inner circle at all! To confirm our findings, we decided we needed a dynamic model that would allow us to experiment with a variety of radii and distances travelled.

We built an interactive model of our problem in Geogebra. You can investigate the model here. Try experimenting with it!

What do you notice?

How far around the circle do you need to travel before the inner distance becomes shorter than the outer distance? What factors (if any) affect this?

What sort of interesting maths puzzles do you see in your everyday life?

**Spoiler Alert!**

If you are travelling around less than ^{1}⁄_{π} of the circle, it is shorter to travel around the outer circle. However, once you are travelling around *more* than ^{1}⁄_{π} of the circle, then it is shorter to take the path along the inner circle. This is always true, no matter what the radius of the inner circle is.