# How to calculate distance run on athletics track

first time poster and definitely no maths expert.

I am trying to solve a basic problem using an athletics track. The total distance around a standard athletics track is 400m:

If you run in the first lane you run 400m, I am trying to work out the formula to estimate the distance run in the 2nd, 3rd, 4th lane etc if they all start at the same point (no staggers)

If I assume that the distance between lane 1 and two is 1m how would I go about calculating?

Any help would be much appreciated! Thanks

• We would have to know the dimensions of the track. The total length is 400m, yes, but do we know the radius of the turn, for example? Jun 24, 2020 at 11:30
• You have 2 half circles (so one circle) and 2 straights. How long is each straight (it looks less than 100m)? Now work out the circumference of your inner circle. The next lanes circle will have a radius 1m wider.
– Paul
Jun 24, 2020 at 11:30
• for this example lets say the straights are exactly 100m each. Does that help? @MattiP. and @Paul?
– SOK
Jun 24, 2020 at 11:32
• 36.50m the radius of the semi circles
– SOK
Jun 24, 2020 at 11:33

Let $$L$$ be the distance of the straight part, and $$R$$ be the radius of the turns. Now, the total length (when you're running on lane 1) is $$400~\text{m}$$, or $$\tag{1} 2L +2 \pi R = 400$$ We also assume that the straights are exactly $$100~\text{m}$$. Then we see that on lane 1, the radius of curvature is $$R = \frac{400-2\cdot 100}{2\pi}\approx 31.83~\text{m}$$.
When moving from the first lane to the next one, the radius $$R$$ increases by 1 meter, and the straight parts remain the same. Therefore, as we can see from Equation (1), the total length increases by $$2\pi$$ meters per every meter that the radius increases.