What a race predictor actually predicts
A race predictor converts one measured performance into the performances at other distances that represent the same level of fitness. It does not predict what you will run next Sunday. It answers a narrower and more useful question: given that you ran 45:00 for 10 km, what would an equally well-executed, equally well-trained half marathon look like?
The distinction matters because everything the model does not know sits in the gap between those two questions. It does not know your weekly mileage, whether you have ever run beyond 15 km, how you handle heat, or whether you can take on fuel while moving. Those factors decide marathon outcomes far more often than raw speed does, and none of them appear in the equation.
Used properly, the prediction is a ceiling and a sanity check. If the model says 1:39 for the half and you are planning 1:32, you are planning to run several minutes beyond your demonstrated fitness. If it says 1:39 and you are planning 1:45 because you want a comfortable long effort, that is a legitimate choice made with the number in hand.
Why the exponent is 1.06 and not 1
If humans could hold one speed for any duration, time would be exactly proportional to distance and the exponent would be 1: a 20:00 5 km would mean a 2:48:48 marathon. Nobody does that, because sustainable speed falls as duration rises. Peter Riegel's model captures the whole of that decline in a single number.
The relationship is a power law, T = a·Db. Take logs and it becomes a straight line: log T = log a + b·log D. Plot world-record times against distance on log–log axes and the points fall remarkably close to a line whose slope is about 1.06 across the endurance range. Riegel published that fit in American Scientist in 1981, and the same exponent has survived four decades of use because it keeps working.
Read the exponent as a fatigue tax. Doubling the distance multiplies the time by 21.06 = 2.0849, so you pay a 4.2% surcharge on pace for each doubling. Quadrupling multiplies it by 41.06 = 4.3469, an 8.7% pace surcharge. Since the ratio is all that matters, the same tax applies whether you go from 5 km to 20 km or from 10 km to 40 km.
The exponent is a population average, and yours is personal. A runner with a strong aerobic base and high mileage typically fits an exponent below 1.06; a fast, low-mileage runner fits one above it. If you have results at two distances, you can solve for your own value: b = ln(T2/T1) ÷ ln(D2/D1). Enter that number in the advanced field and every prediction on the page is recalibrated to you.
The Cameron formula takes a different route to the same destination. Instead of one exponent it uses a fitted function a(x) of distance in miles, so the effective fatigue rate changes with distance rather than staying fixed. The two methods agree closely between about 5 km and the half marathon and separate at the extremes, which is exactly why this page shows both.
Worked example: a 45:00 10 km projected to the half marathon
You ran 45:00 for 10 km and want a realistic half-marathon target. The half marathon is 21.0975 km.
- Put the time in one unit. 45:00 = 45 minutes.
- Form the distance ratio. 21.0975 ÷ 10 = 2.10975.
- Raise it to the exponent. ln(2.10975) = 0.746550; × 1.06 = 0.791343; e0.791343 = 2.206403.
- Multiply by your time. 45 × 2.206403 = 99.288 minutes, which is 1:39:17.
- Convert to pace. 99.288 ÷ 21.0975 = 4.7062 min/km = 4:42 per kilometre, or 4.7062 × 1.609344 = 7.5735 min/mi = 7:34 per mile.
The Cameron formula, run on the same inputs, returns 99.215 minutes — 1:39:13, four seconds faster. Two independent models landing within four seconds over 21 km is about as much agreement as this exercise ever produces, and it is a good sign that 1:39 is the honest number. Take that pace over to the running pace calculator to lay out kilometre splits.
How much to trust the number
Trust it in proportion to three things: how close the two distances are, how recent and well-executed the input race was, and how specifically you have trained for the target.
Distance ratio. Between neighbouring race distances — 5 km to 10 km, 10 km to half — the model is reliable enough to plan splits from. Projecting a marathon from a 5 km stretches one fitted exponent across an eight-fold span, and it assumes your endurance decays at the population-average rate over a range where individual differences are largest. The calculator flags any ratio beyond four in either direction.
Direction of the extrapolation. Predicting down from a long race to a short one is generally the safer direction, because anyone who can complete a marathon has the aerobic base for a 5 km; what they might lack is top-end speed, which makes the short prediction slightly optimistic. Predicting up exposes you to everything the model omits, which is why marathon predictions from short races are the ones runners most often fail to hit.
Quality of the input race. Feed the model a race you actually raced: a certified course, decent conditions, an even effort. A time set on a hilly course, in heat, or in a training block with heavy legs understates your fitness, and every projected time inherits that understatement.
One structural caveat: the model has no idea whether your goal pace is metabolically sustainable for the target duration. Cross-check it against your training zones with the heart rate zones calculator, or against a fitness score derived from race performance with the VDOT calculator, which also generates training paces rather than just an equivalent time.
Equivalent race times from a 5 km result (Riegel, b = 1.06)
| 5 km | 10 km | Half marathon | Marathon (Riegel) | Marathon (Cameron) |
|---|---|---|---|---|
| 15:00 | 31:16 | 1:09:00 | 2:23:52 | 2:26:24 |
| 17:30 | 36:29 | 1:20:30 | 2:47:51 | 2:50:48 |
| 20:00 | 41:42 | 1:32:00 | 3:11:49 | 3:15:11 |
| 22:30 | 46:55 | 1:43:30 | 3:35:48 | 3:39:35 |
| 25:00 | 52:07 | 1:55:00 | 3:59:47 | 4:03:59 |
| 27:30 | 57:20 | 2:06:30 | 4:23:45 | 4:28:23 |
| 30:00 | 1:02:33 | 2:18:00 | 4:47:44 | 4:52:47 |
Generated by running this calculator at each 5 km time. The 10 km column is exactly 5 km × 2^1.06 and the half is exactly 5 km × 4.2195^1.06, so any row can be checked by hand.
Assumptions and limits worth knowing
- One exponent describes every distance. Real fatigue curves bend; a single power law is a straight line on log–log axes. The fit is good over the middle of the endurance range and worst at the sprint end, where anaerobic capacity dominates.
- No training history is in the model. Two runners with identical 10 km times and weekly mileages of 30 km and 100 km get identical marathon predictions. The high-mileage runner will beat theirs and the other will usually miss.
- Course, weather and fuelling are absent. Heat, humidity, wind, elevation gain and a mismanaged fuelling plan can all cost more time over a marathon than the difference between the two formulas on this page.
- The prediction inherits every flaw in the input. A 10 km run as a hard tempo effort rather than an all-out race will project a soft target at every other distance.
- It says nothing about how to get there. An equivalent time is a statement about current fitness, not a training plan. Closing the gap between a predicted and a desired time requires specific work, not a different calculator.
- Very short distances break the assumptions. Below about 1,500 m the limiting system is no longer predominantly aerobic, and both formulas are being used well outside the range they were fitted on.
Find your own exponent from two races
If you have honest results at two distances, solve the power law for b instead of assuming 1.06: b = ln(T2/T1) ÷ ln(D2/D1). A 20:00 5 km and a 1:33:00 half give ln(93/20) ÷ ln(21.0975/5) = 1.5369 ÷ 1.4390 = 1.068. Enter that in the advanced field and the marathon prediction moves from 3:11:49 to 3:14:31. Recompute it after every new race; the value drifts as your training changes.
Riegel, Cameron and the alternatives
Riegel's power law is the oldest and most widely embedded of the equivalent-time models, and it wins on transparency: one exponent, one line of arithmetic, checkable on paper. Cameron's formula, published on the internet running lists in the 1990s, replaces the fixed exponent with a fitted function of distance and tends to be a little more conservative at the marathon end, which is where Riegel is most often accused of optimism.
A third family works from oxygen cost rather than curve fitting. Jack Daniels' VDOT tables convert a race time into an estimated VO2max-equivalent using the measured oxygen cost of running at a given speed and the fraction of maximum you can sustain for a given duration, then convert that score back into times and training paces. It is the most physiologically grounded of the three and the only one that also tells you how fast to run your intervals; the VDOT calculator covers it.
All three assume flat, certified courses and a fully trained athlete. If your target event is a triathlon run leg, remember that you arrive at it with 60–180 minutes of prior work in your legs, so plan that split with the triathlon finish time calculator rather than treating it as a standalone road race. And if the target session is on a treadmill, convert the road pace with the treadmill incline pace calculator before setting the belt.