# Running Lanes and Extra Steps

When going out to your local running track for a workout, you sometimes find that you are allowed to use only certain lanes for training. On any looped track, however, the outer lanes are longer than the inner lanes. That presents a problem for someone using the track for speed workouts.

With that in mind, a runner once submitted the following question to Runner’s World magazine: How can I find the exact distances of all the lanes?

“This is a common question,” veteran runner Elliott A. Weinstein of Baltimore wrote in the April Mathematics Magazine. “. . . the inner lanes are yielded to faster runners by protocol, to slower runners and walkers who ignore or are ignorant of the protocol, and to the entire high school marching band, which just happens to be practicing on the track during your workout and is not bound by the protocol.”

Two coaches replied to the Runner’s World question. One coach had gone out to his 400-meter, eight-lane college track and measured the distances in lanes 4 and 8 (but he didn’t say how). Given these two values, he suggested determining the distances of the remaining lanes by interpolation. The other coach recommended running a lap in each lane while holding the pace steady throughout and noting the time differences.

Weinstein suggested that a mathematical approach would give a much better answer.

A typical 400-meter quadrant track consists of two parallel straightaways connected at the ends by concentric semicircles. For any closed, convex curve, such as a lane divider, the extra distance traveled along a path that is everywhere a distance d away from and outside the curve is simply 2pd.

“Note that this result is independent of the distance around the interior curve,” Weinstein remarked.

For any given track, every lane has the same width. Hence, for a track with lane width w, the extra distance around the track when running in lane n is 2pw(n – 1).

“The beauty of the result is that it holds so generally that it is still true for a [double-bend running track] as well, not to mention an ellipse, a horse track, the letter D . . . or even, say, to a run around a nonconvex lake,” Weinstein says.

The standard lane width for most high school and college outdoor tracks in the United States is 42 inches. This gives 6.70 meters of extra distance per lane per lap. So lane 4 would have a distance of 420 meters and lane 8 a distance of 447 meters, rounded to the nearest meter.

It turns out that the coach’s measurements were off by 1 meter in one case and 2 meters in the other. He might have obtained better results simply by taking the differences in the lanes’ staggered starting marks for an appropriate track event. Even then, he might still have been off because he failed to make the right measurements. Rules specify that a lane’s length is not measured around the outer edge of the lane’s inner boundary but around an undrawn curve, called the measure line, which is 20 centimeters outside that outer edge.

“Apparently,” Weinstein concludes, “it’s safer and easier just to do the math!”