In 1977, engineer and masters runner Peter Riegel published a strikingly simple observation in Runner’s World, later expanded in his 1981 American Scientist paper “Athletic Records and Human Endurance”: race times across distances follow a power law. The formula that bears his name:

T2 = T1 × (D2/D1)^1.06

Take your time T1 at distance D1, multiply by the ratio of the new distance D2 to D1 raised to 1.06, and you get a predicted finish time T2. One constant — 1.06 — carries all of the model’s assumptions about how humans slow down.

What the exponent is really saying

Because time = pace × distance, dividing both sides by D2 shows what happens to your pace:

pace2 / pace1 = (D2/D1)^0.06

In words: every doubling of race distance costs about 4.3% of pace (2^0.06 ≈ 1.043). Go from 5K to 10K and your per-kilometer pace should decay ~4%; from 10K to the marathon — a bit more than two doublings — roughly 9–10%.

Worked example, the one on our calculator’s default: a 50:00 10K predicting a marathon.

• Ratio: 42.195 ÷ 10 = 4.2195
• Factor: 4.2195^1.06 ≈ 4.600
• Prediction: 3000 s × 4.600 ≈ 13,801 s = 3:50:01
• Implied pace: 5:00/km → 5:27/km — an 9% decay, right on the 1.06 curve.

Going the other direction works identically: a 4:00:00 marathon implies a 10K of 14,400 ÷ 4.600 ≈ 3,130 s ≈ 52:10.

Where it earns its keep

The formula is remarkably good for what it is. Between neighboring distances — 5K→10K, 10K→half, half→marathon — and for runners whose training genuinely covers the longer distance, predictions land within a few percent for most people. Coaches use it to set realistic goal times; race calculators use it to translate “I’m a 45-minute 10K runner” into marathon pacing. That’s the mode the race-prediction tab implements.

Where it quietly breaks

Riegel himself was careful about scope; users often aren’t. The known failure modes:

1. Undertrained long jumps. The formula assumes fitness matched to the target. A 20-minute 5K runner with no mileage base will not run the predicted ~3:10 marathon — the model can’t know their longest run is 12 km. Prediction bias: systematically too fast on big upward jumps.
2. Specialist athletes. Elites have flatter personal curves — effectively lower exponents — so marathon-to-5K predictions for a marathon specialist come out too slow, and 5K-to-marathon for a miler comes out too fast. Your personal exponent is only 1.06 on average.
3. Non-flat reality. Hills, heat, altitude, wind and crowding all sit outside the model. A flat-course 10K predicts a hilly marathon optimistically.
4. Distance extremes. Below ~800 m and past the marathon into ultras, other limiters dominate; the power law keeps extrapolating regardless.

This is why our tool shows a window, not a point: ±4% for near-identical distances, widening by about 4 points per doubling gap, capped at ±16%. That band is a heuristic honesty layer, not part of Riegel’s original math — a 10K→marathon prediction deserves ~±12% of humility.

The competition

• Daniels’ VDOT (Jack Daniels & Jimmy Gilbert, 1979) computes equivalent performances from oxygen-cost curves and workout data — more physiology, more tables, similar answers for mid-pack runners.
• Cameron’s model adds a speed-vs-endurance split instead of a single exponent.
• Purdy points score performances against estimated world-class standards rather than extrapolating one result.
• Modern race calculators (and some watches) blend several of these plus your training history — which is the right instinct, since a number is only as honest as its error bars.

The takeaway

Riegel’s formula is the best simple answer to the most-asked question in running — “what could I run at distance X?” Use it for what it is: a center-of-distribution estimate for a runner whose training fits the distance. Then check the window, not just the point — and if you’re planning to run it as a negative split, the predicted time is the ceiling your pacing plan defends, not a promise.

Frequently asked questions

What does the 1.06 exponent actually mean?

It's an empirical fatigue factor. Because T = pace × distance, T2/T1 = (D2/D1)^1.06 means your pace scales as (D2/D1)^0.06 — every doubling of distance slows your pace by about 4.3%. It's not derived from physiology; it's the exponent that best fits real race results across distances.

How accurate is a Riegel prediction?

For well-trained runners between neighboring distances (5K→10K, 10K→half), typically within a few percent. For large jumps (5K→marathon) it systematically over-predicts undertrained runners — the formula assumes your endurance was built for the longer event. We show a ±4–16% window on the calculator that widens with the distance gap for exactly this reason.

Can I use it for ultramarathons or a 400 m?

Outside roughly the 800 m–50 km window the fit degrades: sprint events are limited by different physiology, and ultras add terrain, fueling and hiking that a pure distance ratio can't see. Treat anything beyond the marathon as a rough floor, not a prediction.

Are there better predictors?

Yes — Jack Daniels' VDOT tables derive equivalent performances from oxygen-cost models; the Cameron model and Purdy points fit different curve shapes; modern race calculators blend several. They mostly agree within a percent or two for mid-distance runners, which is a good sign that the honest answer is a range, not a number.