Race Time Predictor Calculator
Race Time Predictor Calculator
Riegel’s endurance equation: a result at one distance turned into a predicted time at another, with the exponent as an input because it is a fitted number and not a constant. A marathon from a 5 km is where it is weakest, and this page shows by how much.
Riegel prediction from a known result
T₂ = T₁ × (D₂ ÷ D₁)^k20 minutes 0 seconds for 5,000 m, predicting 42,195 m at k = 1.06
Formula
Riegel’s own form: t = a × xb, where t is time, x is distance, and a and b are fitted for each activity
- k, the fatigue factor
- FITTED to race records, not derived from physiology. Riegel’s 1981 paper in American Scientist fits it at 1.07732 for world-class men, 1.08283 for world-class women, and 1.05352 to 1.06370 across men’s age groups from over-40 to over-70. The widely quoted 1.06 sits at the bottom of that masters range and is NOT the value his world-class fit produced
- other activities, same equation
- Table 1 also fits swimming at 1.02977 for men, Nordic skiing at 1.01421, race walking at 1.05379, cycling at 1.04834 and speed skating at 1.06017. A lower exponent means endurance falls away more slowly with distance — the clearest evidence that k belongs to a data set and not to a law
- the range it was fitted over
- “about 3.5 to 230 minutes”, with “linearity … lost at times below 3 to 4 minutes” and “I have avoided extrapolation beyond the limits of the endurance range because of the obvious pitfalls”
- D₂ = D₁ is degenerate
- the ratio is 1 and 1 raised to any power is 1, so the prediction returns the known time whatever k is: a useful sanity check on an implementation and a worthless test of the exponent
- what the equation cannot know
- predicting a marathon from a 5 km assumes marathon-specific endurance a 5 km does not measure. The arithmetic here is exact; the model it implements is not
Worked example
20 minutes 0 seconds for 5,000 m, predicting 42,195 m at k = 1.06
20:00 is 1,200 seconds, and the distance ratio is 42,195 ÷ 5,000 = 8.439
8.4391.06 = 9.5911, so the prediction is 1,200 × 9.5911 = 11,509.3 seconds, which splits as 3 hours 11 minutes 49 seconds
At Riegel's own world-class men's fit of 1.07732 the same inputs give 11,942.4 seconds, or 3:19:02 — a divergence of 433.1 seconds, seven minutes thirteen, from a change in the third decimal place of an exponent
The known pace is 240.0 s/km and the predicted pace 272.8, so the equation claims a marathon 32.8 seconds a kilometre slower than the 5 km
Set the distance to predict to 5,000 and the prediction returns 1,200.0 at every exponent, with the divergence at zero — the degenerate case
Go the other way: a 3:00:00 marathon predicts a 5 km of 1,126.0 seconds at 1.06 and 1,085.2 at 1.07732, so extrapolating DOWN the larger exponent gives the FASTER time. The direction flips with the direction of the extrapolation
Riegel’s own fitted exponents, from his 1981 Table 1
| Activity and group | Fitted exponent | Compared with running |
|---|---|---|
| Running, world-class men | 1.07732 | the baseline fit |
| Running, world-class women | 1.08283 | falls away slightly faster |
| Running, men over 40 to over 70 | 1.05352 to 1.06370 | flatter |
| Race walking, men | 1.05379 | flatter than running |
| Cycling, men | 1.04834 | flatter again |
| Speed skating, men | 1.06017 | close to the quoted 1.06 |
| Swimming, men | 1.02977 | much flatter |
| Nordic skiing, men | 1.01421 | the flattest in his table |
| Commonly quoted for running | 1.06 | not from his world-class fit |
The same 5 km, predicted two ways
| Known 5 km | Marathon at k = 1.06 | Marathon at k = 1.07732 | Divergence |
|---|---|---|---|
| 15:00 | 2:23:52 | 2:29:17 | 5:25 |
| 18:00 | 2:52:38 | 2:59:08 | 6:30 |
| 20:00 | 3:11:49 | 3:19:02 | 7:13 |
| 22:00 | 3:31:00 | 3:38:57 | 7:57 |
| 25:00 | 3:59:46 | 4:08:47 | 9:01 |
| 30:00 | 4:47:43 | 4:58:33 | 10:50 |
An exact calculation over a fitted model
This page is about athletics — track and field, and road running — whose competition rules, technical rules and scoring tables are published by World Athletics, the body that was the IAAF until 2019. Where a figure below belongs to a particular edition of those rules, the edition is named beside it, because World Athletics revises them and a number quoted without its edition will be wrong for somebody.
Riegel’s endurance equation says time rises as a power of distance: t = a × xb, with a and b fitted for each activity. Rearranged between two distances it loses the constant and becomes the form every race predictor uses, T₂ = T₁ × (D₂ ÷ D₁)k. Peter Riegel published it in Runner’s World in August 1977 and the thesis behind it in American Scientist in May 1981.
A note on how a time is shown here. The calculator on this page is numeric: it has no text type, so it cannot print a colon and there is no “4:32” for it to show. The headline is therefore in seconds, and the whole minutes and the remaining seconds are given as separate figures beneath it — 272 seconds per kilometre, then 4 and 32. The split is taken on the rounded whole second rather than on the raw value, which matters more than it looks: a pace of 239.6 seconds per kilometre split before rounding reads “3 minutes and 60 seconds”, and rounding first makes it 4 minutes and 0 seconds.
The exponent is where the honesty has to be. It was fitted to record performances, not derived from anything, and Riegel’s own Table 1 fits it at 1.07732 for world-class men and 1.08283 for world-class women — not the 1.06 almost every calculator uses, which sits at the bottom of the range he fitted for men’s masters age groups. The same table fits swimming at 1.03 and Nordic skiing at 1.014, which is the clearest demonstration that the number belongs to a data set rather than to endurance. So it is an input here, and the panel prints the prediction at Riegel’s own world-class figure beside whatever you choose: on a 20-minute 5 km extrapolated to a marathon the two differ by seven minutes thirteen seconds.
Riegel stated his own limits: the equation was fitted over “about 3.5 to 230 minutes”, “linearity is lost at times below 3 to 4 minutes”, he had “avoided extrapolation beyond the limits of the endurance range because of the obvious pitfalls”, and “the endurance equation is not set forth as a fundamental theory of human locomotion”. A marathon predicted from a 5 km is a ratio of more than eight, and what it cannot know is whether the longer distance has been trained for: a 5 km says nothing about marathon-specific endurance. That gap is what the published criticism of the formula is about and why other models exist. This page does not tell you how to close it — there is no training advice here, only arithmetic and its limits.
Where the line runs between this page and the health section: health owns the athlete’s body and this page owns the race. A figure derived from a person — aerobic capacity, a heart-rate zone, a body composition — is there, at the VO2 max calculator and the target heart-rate zones calculator. A figure derived from a clock and a distance is here. Nothing on this page prescribes a training pace or a heart rate, or makes any physiological claim.
To turn a prediction into a race plan, the race split times calculator takes the pace and returns the splits. For a result compared against what is typical at a given age rather than at a given distance, the age-graded performance calculator uses published factor tables instead of a fitted exponent.
Frequently asked questions
What is the Riegel formula?
T₂ = T₁ × (D₂ ÷ D₁)^k, where T₁ is a known time over distance D₁ and k is a fitted exponent. It comes from Peter Riegel’s endurance equation t = a × x^b, in Runner’s World in August 1977 and American Scientist 69(3) in May 1981.
Is the Riegel exponent really 1.06?
1.06 is the value the formula is usually quoted with, but it is not the one Riegel’s own world-class running fit produced. His 1981 Table 1 gives 1.07732 for world-class men and 1.08283 for world-class women, and 1.05352 to 1.06370 across men’s masters age groups. The exponent is an input on this page because it is a fit, not a constant.
Can I predict a marathon from a 5 km?
The arithmetic will do it; treat the answer with care. Riegel fitted the equation over times of roughly 3.5 to 230 minutes and wrote that he avoided extrapolation beyond that range “because of the obvious pitfalls”. A 5 km does not measure marathon-specific endurance, so the prediction assumes something it cannot see. The page prints the divergence between two published exponents so you can judge how soft the number is: over seven minutes on a 20-minute 5 km.
Why does the prediction return the known time when the distances match?
Because the distance ratio is 1 and 1 raised to any power is 1. It is a good check that the formula is wired correctly and a useless check on the exponent, which cancels completely there.
Does a bigger exponent always predict a slower time?
Only extrapolating upwards. Predicting a marathon from a 5 km, a larger exponent gives a slower marathon; predicting a 5 km from a marathon it gives a faster 5 km, because the ratio is now less than one.
Related calculators
References
- Riegel, Peter S. “Athletic Records and Human Endurance.” American Scientist 69, no. 3 (May–June 1981): 285–290. The endurance equation “t = axᵇ, where t = time, x = distance, and a and b are unique for each activity”. Table 1 fits the fatigue factor at 1.07732 for world-class men and 1.08283 for world-class women in running, 1.05352 to 1.06370 across men’s age groups from over-40 to over-70, and a range of lower values for swimming, skiing, walking, cycling and skating; the text says “World-class runners, men and women, have an identical fatigue factor of 1.08”. The endurance range is “about 3.5 to 230 minutes”, “linearity is lost at times below 3 to 4 minutes”, “the endurance equation is not set forth as a fundamental theory of human locomotion”, and “I have avoided extrapolation beyond the limits of the endurance range because of the obvious pitfalls”. Read 12 Oct 2026 from the scanned copy hosted by Northern Kentucky University; about 35,000 characters including JSTOR front matter, with Table 1 readable once its split columns are realigned and the five figures unreadable OCR.
- Wikipedia. Peter Riegel (read 12 Oct 2026; about 5,000 characters). Gives the earlier publication as Riegel, P. S., “Time Predicting”, Runner’s World, August 1977, and records that the formula “is most commonly quoted with” an exponent of 1.06 and that some sites have modified it because 1.06 “results in seemingly unachievable predictions for longer distances”. Cited for the 1.06 attribution and the published criticism of it, not for the formula.
- World Athletics. “Marathon” discipline page, worldathletics.org (read 12 Oct 2026; about 20,000 characters, much of it navigation). “Athletes run a distance of 26 miles and 385 yards (42.195km)”, and “this distance was standardised at 26 miles 385 yards (42.195km) in 1921”.