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## Thursday, May 27, 2010

### Just How Long Is That Quarter Mile Track Anyway?

This morning I got up early and came to school to run around the track. (I know, right? Nobody was chasing me or anything.) And I know it's a quarter-mile track, but of course the inside lane is much shorter than the outside lane. So I started wondering, what is a quarter mile? The inside, the outside, somewhere in the middle? I'm getting this down now so I remember to revisit it in Geometry next year when we do all those composite perimeters and areas.

Here is our track from Google Earth:

I'll ask the kids to decide what to measure.

If you try to measure in miles, the diameters of the semicircular ends only differ by a hundredth of a mile. So I measured in feet for more precision.

I'd love to give the kids a printout and have them do the measuring, but measuring in centimeters or inches and scaling it up might be more trouble than its worth. Maybe if I find a few more interesting questions like this, we can have a computer day and they can use the GE measuring tools. Times like these I envy you 1:1 schools.

For the outside track, rounding to the nearest foot I get 1444 feet or 0.273 miles and for the inside track, I get 1320 feet which is exactly 0.25 miles. Can anyone verify that? Track stars?

1. Every track runner should know that it's only an exact quarter mile on the inside of the track. That's why they have to offset the starts for track races, so everyone runs the same distance as the guy on the inside.
The real question is why do joggers choose to use the inside track? Always noticed that, never understood.

2. Trackstar #2 verifying. The inside of the track is exactly a quarter mile. The outer lanes are staggered based on the number of turns that a runner will take while in those lanes. Distance runners start in alleys (a group of lanes) and are expect to flatten into the lowest lane of that alley until the break mark (where they can cut in to the inside lane).

@Max -
It's been my experience that joggers like to count lap numbers for distance instead of using a pedometer or something of that nature. A few places (colleges in particular) will not allow joggers in the inside 3 lanes to avoid excess wear on the track.

3. Thanks! So gratifying when math works.

4. Maybe you could throw this picture into Geometer's Sketchpad or Geogebra for the measuring.

5. Another neat observation: each time you move outward one lane (a distance of 4', say), your lap length increases by 2*π*4, or approximately 25 feet. This has nothing to do with which lane you started in, the shape of the track, or the length of the track.

6. In high school, they told us that we should estimate 7 meters extra length per lane. I remember several times arguing to no avail that the lane width (which is far from standard) was extremely relevant.

Anyway, I can't wait to figure out somewhere to use this.

7. I love it!

And it sparked a few questions of my own (some that have already been addressed here).

1. If you know the width of each lane, where should each runner start to ensure that everyone runs the same distance (you'd also need to know the distance of the race and assume that runners stay in their lanes)?

2. Not really a math question, but..how much harder is it to run in a tight circle (ie the inside lane)?

3. An old classic...There is a rope that extends around the entire earth at the equator. That's why there's a line there :) If I wanted to make it so this rope was a foot off the ground everywhere, how much extra rope would I need?

8. @Maestro I guess that makes sense, but it can't help the wear on the track, which is what bothers me.

@Avery I would say it isn't much harder to run on the inside, but for all of the distances that are run on the track it can make a relatively large difference in time. One of the things I've pondered on those endless loops is whose job it is to paint those lines on new tracks that tell you where to start, if they have to calculate before they paint. Probably not something most painters have to worry about!

9. This is a great case of accuracy vs. precision. While Google Earth feels comfortable giving you very precise measurements, in my experience measurements in Google Earth usually aren't accurate to be talking about distances in terms of feet

10. If you know how high the cross bar on the football goal post are, can you use some shadow lengths to get an estimate of the heights of the light towers on each side of the stadium? Or perhaps the height of the building in the lower right corner?

11. The inside line of the inside lane should measure out to be exactly 400 meters. Sorry, English measurements lose out here. On older tracks, the inside lane is exactly 440 yards. So on a 440 yard track, a runner in lane 1 will run exactly 1 mile. On a 400 meter track, the runner will only run 1600 meters, which is 31 feet short of a mile.

Each lane measures 1.22 meters (I'm kinda sure), so on a standard 8 lane track, the inside of lane 8 is 8.54 meters away from the inside of lane 1.

Once you figure out how much further you would run in lane 8, take a field trip to the track and have them measure the stagger marks at the start line (probably the yellow ones, but they should be marked).

12. yep! inside line of the inside lane! that's why they do stagger starts! very cool math, k8! :)

13. What an awesome practical use of mathematics!!

Instead of using Google Earth, have you thought about taking your class on a "field trip" to the track to actually measure and then find the distances of each lane? It sounds like a great hands-on lesson about perimeter and circumference. You can even do more if you ask about the area of each lane and the infield of the track and so on.

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