Pace, time and distance: the running math, plus Riegel's race estimates

Three quantities, one equation

Pace is time per unit of distance. If you know the time and the distance, divide. If you know the pace and the distance, multiply to get the time. If you know the time and the pace, divide time by pace to get the distance.

The only trap is the clock format. Pace is written in minutes and seconds, such as 5:00, so convert to seconds or decimal minutes before doing arithmetic, then convert back.

The split table applies the same rule one step at a time. At an even 5:00 per km, the clock reads 5:00 at 1 km, 25:00 at 5 km and 50:00 at 10 km. Each row is simply the pace multiplied by the distance covered so far. Real races rarely run perfectly even, so splits are a reference line to compare against, not a forecast of each kilometer.

Kilometers, miles and speed

One mile is 1.609344 kilometers. A pace per mile is therefore the pace per kilometer multiplied by 1.609344. Going the other way, divide.

Speed is the same information turned around. A pace of 5:00 per kilometer means 12 kilometers in 60 minutes, so 12 km/h. Divide by 1.609344 for miles per hour.

Standard race distances

The quick buttons fill in the common road distances. World Athletics gives the half marathon as 21.0975 km (13.1094 miles), exactly half a marathon, so the marathon is 42.195 km.

• ‘5K’: 5 km.
• ‘10K’: 10 km.
• ‘Half marathon’: 21.0975 km.
• ‘Marathon’: 42.195 km.

Riegel's race time formula

Peter Riegel was an American research engineer. He first published a time-prediction method in Runner's World in August 1977. In 1981 he set out the underlying pattern in American Scientist, in the article Athletic Records and Human Endurance.

In that article, world-record times fit a simple power law, t = a × d^b, where b is above 1. An exponent above 1 means each extra kilometer costs a little more time than the last, because pace slows as distance grows. Riegel called b the fatigue factor.

The version used by most race calculators fixes the exponent at 1.06, which gives T2 = T1 × (D2 ÷ D1)^1.06. Running calculator sites describe typical values from 1.06 to 1.10.

Limits of the estimate

The 1981 article fitted records, not individual runners. Its exponent for men's running world records was about 1.077 and for women's about 1.083, a little steeper than 1.06. For men aged 40 to 70, the fatigue factor was 1.05 to 1.06. So the fixed 1.06 sits at the optimistic end for record-level data.

Riegel limited the fit to an endurance range of efforts between about 3.5 and 230 minutes. He wrote that he avoided extrapolation beyond those limits because of the obvious pitfalls. A 3:50 marathon sits right at the upper edge; slower marathon estimates go past it.

He also described the equation as only a simple description of what is observed, not a fundamental theory. He noted that runners used it with success to predict their future times. Training still matters. Recent training and how specific it was to the distance matter too.