How far is 1 lap in lane 8 on an outdoor 400m track

You're right -- if you're calculating the DIFFERENCE between two lanes, only the turns matter, because the length of the straightaways is the same for each lane. Your calculation here is basically the same as mine, except I tried to use the stagger (which I thought was 10m for two turns for lane 2 over lane 1, but I'm told is 6m for two turns) directly, only using the circumference formula to show that the relationship was indeed linear, because I had no idea what the width of a lane was.

All this is assuming, of course, that the turns are circular (on a standard track, I think they are.)

438 meters

The Life of Pi does an outstanding job of mocking organized religion, which deserves some mocking from time to time.

D_Gordon wrote:

All this is assuming, of course, that the turns are circular (on a standard track, I think they are.)

I repeat:

The turns don't have to be circular!

The track can be any shape you want. It can turn left for a while, it can turn right for a while, it can do a little loop-the-loop if you want. As long as you end up turning left 360 degrees more than you turned right, you will have traveled

2*PI*(width of lane)

more in lane 2 than in lane 1.

Hey buddy,

get back to grade school. They do need to be circular for your formula to work. Think about it a little...

Actually a pretty accurate formula that i found a while back in a book is:

COS(pie) / 3*(lane 8 diameter - lane 1 diameter)^3 + 2* sqrt(track length) + 0.774*(lane width)

just plug those numbers in and you'll get the result you're looking for

Pete wrote:

Hey buddy,

get back to grade school. They do need to be circular for your formula to work. Think about it a little...

I did think about it a little Pete. Please see my post on page 1 of this thread and explain to me what's wrong with my argument.

Assuming that the question was asked for practical reasons and not out of academic interest, just measure backwards from the start of the 400 meter run to the common finish. I've never cared enough to measure it more accurately than a number of seconds per lap at a given pace. If it takes me 99 seconds to run a lap, and 90 seconds to run from the 400 meter start to the common finish, I have a good enough idea of how far I'm running.

The turns can be elliptical but noncircular - then the formula for circumference of a circle wouldn't apply. I have no idea why they would be, and I've never heard of a track with such turns, so this isn't important, but yes, it does matter. If you know the length of the one-turn stagger (say, the difference between the 800m starting line in lane one and that in lane two) you can calculate it the same way, though, because the relationship would still be linear.

Again, this is all academic, as I've never heard of a track with elliptical but noncircular turns.

Nobody seems to be really sure on this one yet ????

guytorrey wrote:

malmo, where are you!

you know EVERYTHING! (i'm in middle school w/pimples and think alan weebb invented the mile run)

Yup. I do. So far the best we've got here, including a few engineers ALMOST got the question correct. One variable everyone here overlooked: Curb or no curb? Without an inside curb everyone here got the question right. With curb, BZZZZZZ, close. (Explanation below)

2(Pi)W(Lanes-1) is the difference.

No need to confuse with the formulae, just look at it from the broadest perspective. There could be two extremes of track design (this is for arguments sake only to illustrate the point) 1) a track with 200m straights, with lanes drawn around a center point at each end, and 2) a track with no straights that is all circle.

In either extreme every single one of you could calculate the difference between the inside lane and each subsequent lane. Using a standard 42 inch (1.067m) lane width the first extreme example would work out to:

2*Pi*1.067*(2-1)=6.70

So the track that we designed with ALL straight-away shows a difference of 6.7 meters per lane. Lane eight = 46.93 meters.

Let's look at the other extreme: the circular track. The radius of a 400 meter circular track would be 63.66 meters. To calculate the length of lane two:

2*Pi*(63.66+1.067)=406.70

Again the difference is 6.70 meters per lane.

No matter what the topography of the track the results will always be the same. Even if the radius changes throughout the turn.

THE CURB changes everything. A track without a curb is measured 20cm from the line, and each and every lane is measured that way. If, in fact, there is a curb, the inside lane is measured 30cm from the curb. Since the lane widths remain constant, that 10cm difference in measurement works out to .628 meters per lap. In other words from "curb to lane2 line" the difference will be 7.33 meters, then each subsequent lane the difference is back to 6.7 meters.

Lane eight will be 47.56 meters longer per lap than lane one if there is a curb. 46.93 meters with no curb.

So, if I want to run miles on a standard high school 400m track (with no curb) ... AND get my 440y splits at the same spot on each lap ... I would run in the outside two-thirds of lane one.

I would like Jason to take a stab at this problem, after all he is the guy with a maths degree. The third poster I think hit the nail on the head.

I'm taking 4 upper level math courses this semester, so I'd be pretty ashamed to mess up on something a 6th grader should be able to figure out.

The people who said the formula 2*pi*lane width for each lane beyond 1 are right. However, it does matter if the track is triangular or square, because in that case the lane lengths CAN'T be uniform everywhere--the necessary condition is for lane lengths to be the same everywhere, if that's the case, it doesn't matter how uncircular or long the track is, it can even do an S shape in places.

D_Gordon wrote:

The turns can be elliptical but noncircular - then the formula for circumference of a circle wouldn't apply.

The formula does work for an elliptical track and you can prove it to yourself by going to

Type in 10 for the long axis and 8 for the short axis, hit calculate and write down the result. Now type in 8 for the long axis and 6 for the short axis and you will find that the result is 2*PI less than the previous result.

As others have now posted the formula holds for any shape track as long as the lanes are a uniform width. A "square" track will work as long as you round the corners to maintain the same lane width.

Kartelite, dukerdog is right, it doesn't matter what the shape of the turns are, just as long as you make a 360 degree turn the formula holds. The track doesn't even have to be symmetrical: if you have an ellipse at on end and a semi-circle at the other the formula will hold. Straights and/or curves are different lengths? Doesn't matter. As long as the lane width is constant and the track length is constant, the only other "variable" is Pi, which of course, is not a variable at all.

Don't take my word on it. Construct imaginary tracks (using any number of the possibilities suggested above), run the calculations, and you'll see: 42 inch lane, 400m track, Pi is constant, 8 lanes, you can only have one answer: 46.93 meters difference (if the inside lane is without a curb).

It doesn't even matter if a drunken sailor designed the curves - the answer remains the same.

While I'm at it, dukerdog, it's Pete who hasn't thought this thing all the way through. You are correct (which of course you already knew) that the track could even make right turns and loop-to-loops like a slot-car track. As long as you make a 360 degree journey and none of the other variables (lane width, track length, Pi) has changed the answer will always be:

2*Pi*W*(L-1)

dude, just run in lane 1

Actually Malmo, the formula I gave:

len diff = 2*pi*diff_between_lanes

is correct even if there's a curb. It requires that the individual knows how to measure the distance properly, not simply lane_width(laneX-laneY) ;-)

As far as shape goes, it does matter in the extreme. The curves do need to be constructed along an arc (normal radius). That's where Pi comes in. Imagine two tracks designed with just straights (lane one is simply running around two pylons 200 meters apart). Now, how do you create the other lanes? Normally, you'd just draw an arc (radius) using the pylon as the origin, in which case the above formulas work. OTOH, what if you constructed the track in a "squared-off" manner? In this second case the outside lane would be constructed of 3 straights, and as the curve of the first track would be circumscribed within the second, the second must be longer. This is largely an academic argument as no sane person would build a track this way, BUT sometimes insane people do build tracks! I bring this up because in high school I remember running on a track that was built around the school's baseall field and which consisted of four straights with hard 90 degree corners! I have no idea how they measured the outside lanes. It was the craziest track I ever saw, and fortunately, I only had to run there once!