Methodology, Constants and Sources

Quick answer:pace is time divided by distance, one mile is exactly 1.609344 km, a marathon is exactly 42.195 km, and race-time equivalents use Riegel's formula t2 = t1 x (d2 / d1) ^ 1.06. Times are computed in seconds at full floating-point precision and rounded only when displayed. Everything below sets out where those figures come from and where each one stops being reliable.

Constants

Formulas

Pace. pace = time / distance. Stored internally as seconds per kilometre. A min/mile pace is converted to seconds per kilometre by dividing by 1.609344 before any other arithmetic happens, so switching the display unit never changes a result.

Speed.kph = 3600 / (seconds per km). mph = kph / 1.609344. The two are always derived from the same underlying pace rather than from each other's rounded display value.

Race-time equivalents.t2 = t1 x (d2 / d1) ^ 1.06, where t1 is a known time over distance d1 and t2 the predicted time over d2. This is Riegel's formula and it is the model behind the race time predictor, the equivalent-times table on the marathon pace chart and the equivalents on every goal-time page.

Even splits. Each segment gets total time divided by total distance. The final partial segment of a marathon, the 195 metres past the 42 km marker, is included, so the last row of every split table lands exactly on the goal time rather than near it.

Negative splits. The splits calculator ramps segment pace linearly from 3 percent slower than average at the start to 3 percent faster at the finish, then rescales so the total still matches the goal exactly. That 3 percent is a presentation choice, not a published finding, and it is stated here so nobody mistakes it for one.

Rounding

All arithmetic runs on seconds as floating-point numbers. Rounding happens once, at display time: times and paces to the nearest whole second, speeds to one decimal place. That means a displayed pace multiplied back out by hand can differ from the displayed finish time by up to a second. The underlying value does not.

Where these models stop being reliable

Riegel's formula flatters long-distance predictions made from short races. The formula was fitted to record performances, that is, to athletes trained for both the source and the target distance. A runner whose longest run is 10 km has the speed the formula reads but not the endurance it assumes, so a marathon predicted from a 5K result is routinely optimistic. Predicting downwards, from a marathon or half marathon to a shorter race, is the more trustworthy direction. Treat any marathon figure predicted from a 5K as the fastest end of a range rather than as a target.

The formula has a validity range. Riegel presented it for endurance events rather than for sprints or for multi-day ultras, and it degrades at both ends. PaceLab does not publish predictions beyond the marathon for that reason. TODO: verifieer the exact duration bounds Riegel states in the 1981 paper before quoting a numeric range here.

The 1 percent treadmill grade is an average, not your number. Jones and Doust measured the grade at which treadmill running matched the energy cost of outdoor running, and found 1 percent to be the best fit at the faster speeds they tested. It is a group average from a laboratory study, and belt calibration, wind, and running economy all move it. The treadmill pace converter therefore converts speed to pace as a direct arithmetic conversion and presents the grade question separately.

GPS distance and course distance are not the same measurement. Certified road courses are measured along the shortest legal route with a calibrated bicycle, and a small safety factor is applied so that a course can never come out short. A runner who does not hold that shortest line, which is everyone in a crowded field, covers more ground than the certificate says. That is why a watch commonly reads over 42.195 km at a marathon finish, and it is a measurement difference rather than a fault in either device.

What PaceLab deliberately does not calculate

Heart-rate zones. The familiar estimate of maximum heart rate as 220 minus age is a population regression with a large spread around it, and later work proposed different coefficients: Tanaka, Monahan and Seals (2001) put maximum heart rate at roughly 208 minus 0.7 times age, and reported substantial individual variation around that line too. Either way the output is a rough population estimate, not your maximum, and building a personal training zone on it can be badly wrong in both directions. PaceLab does not publish heart-rate zones and does not give medical advice. If you want zones, get them from a measured test rather than from an age formula.

Named training-pace systems. Several widely used training-pace tables are published works with named authorship and are not free to reproduce. PaceLab does not reprint them and does not approximate them with invented coefficients. Where the site discusses training pace at all, it does so in terms that follow from the pace arithmetic on this page.

Sources

1. Riegel, Peter S. (1981). Athletic Records and Human Endurance. American Scientist, 69(3), 285-290. PubMed record.
2. Jones, Andrew M. and Doust, Jonathan H. (1996). A 1% treadmill grade most accurately reflects the energetic cost of outdoor running. Journal of Sports Sciences, 14(4), 321-327. PubMed record.
3. Tanaka, Hirofumi, Monahan, Kevin D. and Seals, Douglas R. (2001). Age-predicted maximal heart rate revisited. Journal of the American College of Cardiology, 37(1), 153-156. PubMed record.
4. World Athletics. Competition and Technical Rules, road races. Book of rules. Used for the marathon and half marathon distances.
5. National Bureau of Standards (1959). Refinement of values for the yard and the pound. Federal Register notice defining the international yard as exactly 0.9144 metres, from which one mile is exactly 1.609344 km.

Reporting an error

If a number on this site disagrees with your own arithmetic, that is worth knowing about. The formulas above are all you need to check any of it by hand. Send the page, the input and the figure you expected via contact and it gets checked against the generator.

Last reviewed 2026-08-05. Formulas, constants and sources: methodology. Changes to any calculation are logged in the changelog.