How Do Track Runners Use Geometry?

Track runners run on what is called an "oval" track, though it actually consists of two semicircles connected by two straightaways. The track usually has six to 10 lanes (an Olympic track has eight), and the innermost lane typically measures 400 meters all the way around. (See Reference 2.) Due to the rules of geometry, though, each lane after the innermost one is successively longer than the lane closer to the inside, leaving an opportunity for track runners to use geometry to their advantage.

Explanation

If you had to walk around a trash can and then around a building, which trip would take longer? This is an exaggerated example, but the idea is the same: track runners in the outer lane of the track have a longer trip to take when everyone starts on the same line. It is an issue of perimeter in geometry: the distance around the edge of a shape. The "ovals" on the inside of the track are smaller than the "ovals" on the outside, so the inside perimeters are less than the outside ones.

Staggered Starts

To make up for this problem, a track has separate, staggered starting lines for each runner for the 400-meter and 200-meter races. Both races start on a curve on the track. Runners on the 200-meter course must run around one bend of the track and down a straightaway, tracing the shape of a "J" as they go. Runners of the 400-meter race run a full circuit around the oval track. Because of the extra distance included in the outer lanes, runners' starting points are staggered so that runners' in each of the outer lanes start the race further down the curve than the person in the inner lane next to them.

Why This Works

The circumference of a circle is equal to two times pi times the radius (C = 2πr). Since each lane on an Olympic field is 1.22m wide, the lane line on each runner's left as they start the race is part of a semicircle with a radius 1.22m longer than the person next to them. As the radius gets bigger, so does the circumference. The staggered starting lines compensate for this added circumference by placing each runner 3.14 meters ahead of the runner on the inner lane next to them in the 200-meter race, and 6.28 meters ahead in the 400-meter race (since the latter runners have two curves to run around).

Distance Running

In the mile and the two-mile races, runners are not assigned lanes, and staggered lines are often ignored in non-Olympic events. In these races, everyone starts on the same line and must run four or eight laps. The best position to be in at the start of these races is in the innermost lane and on the starting line; the person there will inevitably run the shortest distance. Everyone else must spend precious steps getting into the inner lane as quickly as possible, while still passing competitors. (In the Olympics, a carefully measured, curved line is drawn across the whole track so that no matter where runners start, they are the same distance from the finish.)

Passing

In these longer races, once you step out of the innermost lane, you are extending the distance you are running, but this usually must be done to pass competitors, because everyone wants to be in the inner lane. It is best to pass on a straightaway, since the outer curves are actually the cause of the extended outer-lane length. You minimize the added distance if you can pass a competitor on a straightaway and get back into the inner lane before the next curve. Using this geometric strategy, a distance runner can ensure the fastest finish time possible.

References

• Track and field Wikipedia
• IAAF Track and Field Regulations