Running Pace and Predicted Race Time

From a distance and a time, finds the pace per kilometre and the speed. It also predicts the time for a different distance with Riegel's formula, T₂ = T₁ × (D₂ ÷ D₁)^1.06.

Pace is how long it takes to cover one kilometre. Divide the time by the distance. For speed instead, divide the distance by the time.

P = \dfrac{T}{D}, \quad v = \dfrac{D}{T} \times 60

$P$ is the pace in minutes per kilometre
$T$ is the time in minutes
$D$ is the distance in kilometres
$v$ is the speed in kilometres per hour

To predict the time for a different distance, use Riegel's formula.

T_2 = T_1 \times \left(\dfrac{D_2}{D_1}\right)^{1.06}

$T_1$ and $D_1$ are the time and distance actually run
$T_2$ and $D_2$ are the time and distance being predicted

What the exponent 1.06 means

This is the whole idea of the formula. If the exponent were 1, time would be exactly proportional to distance, which would mean holding the same pace no matter how far you go.

Runners cannot do that. Pace falls off as distance grows, and 1.06, just a shade above 1, is what encodes the fall-off. That 0.06 does real work: double the distance and the time is not twice as long but 2.08 times, and quadruple it and the time is 4.35 times. The gap widens the further apart the two distances are.

Example

Suppose you run 10 km in 50 minutes. The pace is $50 \div 10$ , exactly 5 minutes per kilometre, and the speed is $10 \div 50 \times 60$ , or 12 km/h.

Now predict a marathon of 42.195 km. The distance ratio is $42.195 \div 10$ , or 4.2195. At the same pace that would be $50 \times 4.2195$ , about 211 minutes, but raising 4.2195 to the power 1.06 gives 4.6, so the prediction is $50 \times 4.6$ , about 230 minutes, or 3 hours 50 minutes and 1 second.

That works out at $230 \div 42.195$ , about 5 minutes 27 seconds per kilometre, 27 seconds slower than the 10 km pace. Planning a marathon at your 10 km pace is how the second half turns painful.

Points to watch

The further apart the two distances, the less reliable the prediction. Extrapolating from 1500 m to a marathon produces a meaningless number. Roughly two to three times the known distance is a sensible limit.

The formula assumes you have trained for the distance. If you can run 10 km but have never raced a marathon, the fade in the closing kilometres is not in the formula, and the real time will be slower.

Heat, wind and hills are not in it either. The same fitness produces slower times in worse conditions.