# Length of Outer Running Track Lanes

Running one round on an outer lane in a stadium’s running track takes longer. The radius of the curve is larger and so is the overall distance of one round. But what exactly is the difference in length? Short answer: $\Delta s=7.037m$ per round.

The IAAF Track and Field Facilities Manual[1] specifies the running track as follows:

The 400m Standard Track comprises 2 semicircles, each with a radius of 36.50m, which are joined by two straights, each 84.39m in length […]. The inner edge of the track is 398.116m in length […]. This length for the inner edge gives a length of 400.001m […] for the theoretical line of running at a distance of 0.30m from the kerb. […] The length of each of the other lanes is measured along a theoretical line of running 0.20m from the outer edge of the adjacent inside lane […]. All lanes have a width of 1.22m ± 0.01m.

From this information the following equation can be derived, whereas $s$ is the length of the theoretical line of running and $n$ the number of the lane (with $n=1$ for the innermost, $n=2$ for the second lane, and so on):

$$s(n) = \begin{cases} \big(36.50m+0.3m\big)\cdot 2\cdot \pi+84.39m\cdot 2, & \text{if n=1} \\ \big( 36.50m+1.22m\cdot (n-1)+0.2m \big) \cdot 2\cdot \pi+84.39m\cdot 2, & \text{if n\ge2} \end{cases}$$

### Track length (one round)

1. 400.001m
2. 407.038m ($\Delta s=7.037m$)
3. 414.704m ($\Delta s=14.703m$)
4. 422.369m ($\Delta s=22.368m$)
5. 430.035m ($\Delta s=30.034m$)
6. 437.700m ($\Delta s=37.699m$)
7. 445.366m ($\Delta s=45.365m$)
8. 453.031m ($\Delta s=53.030m$)
9. 460.697m ($\Delta s=60.696m$)

### Sources

[1] Specification and quote: IAAF Track and Field Facilities Manual 2008 Edition
[2] Running track image (cropped): Track_and_field_stadium.jpg

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Applied Scientist with Zalando and Founder of Denk Development. Interested in data science, software engineering, math, microcontrollers, and sports.
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