How PaceCalc calculates

Pace, time and distance

Pace is time divided by distance. Time is pace multiplied by distance, and distance is time divided by pace. The pace calculator solves whichever one you leave empty.

Even splits assume the same pace for every mile or kilometer, so the time at each marker is the pace multiplied by the distance covered so far.

Units and race distances

One mile is exactly 1.609344 kilometers, the international mile (NIST Special Publication 811, Appendix B).

A marathon is 42.195 kilometers and a half marathon is 21.0975 kilometers, the distances set in World Athletics' competition rules for road races. A 5K is 5 kilometers and a 10K is 10 kilometers.

Pace charts

Each row of a pace chart is one pace. The marathon and half marathon charts divide each finish time by the race distance; the running pace chart starts from a pace per mile; the treadmill chart turns speed into pace as 60 ÷ mph minutes per mile. Every other column is that pace multiplied by a distance, using the same unit definitions and rounding as the calculators.

Split times assume an even pace from start to finish. Race courses are measured along the shortest route a runner could take (World Athletics Technical Rules, Rule 55.3), so a GPS watch, which records the line you actually ran, usually shows a little more than the official distance.

Treadmill incline: the 1% suggestion comes from Jones AM, Doust JH (1996), "A 1% treadmill grade most accurately reflects the energetic cost of outdoor running", Journal of Sports Sciences 14(4):321–327. It tested speeds from about 6.5 to 11 mph; at the slower speeds tested, 0% and 1% were both close to outdoor running.

Split plans

The split calculator divides the race into two equal halves. With a negative split of x%, each mile or kilometer in the second half takes (100 − x)% of the time of one in the first half; with a positive split, (100 + x)%. The first half therefore takes the goal time ÷ (2 − x/100) for a negative split, or ÷ (2 + x/100) for a positive one, and the second half takes the rest. An even split is the goal time ÷ 2 for each half.

The pace changes exactly at halfway, so the split that contains halfway mixes the two paces. Each split is rounded to the nearest second on its own, so adding up the splits shown can differ from the elapsed column by a second or two; the elapsed times are calculated exactly and only rounded for display.

VO2 max

The race method uses Daniels J, Gilbert J (1979), Oxygen Power: Performance Tables for Distance Runners. Running at v meters per minute costs −4.60 + 0.182258v + 0.000104v² ml of oxygen per kg per minute, and the fraction of VO2 max a runner can hold for t minutes is 0.8 + 0.1894393e^(−0.012778t) + 0.2989558e^(−0.1932605t). VDOT is the oxygen cost of the race pace divided by that fraction: an effective VO2 max from performance, not a lab measurement. The equivalent race times are the times at other distances that give the same VDOT, found by bisection.

The 12-minute run uses Cooper KH (1968), "A means of assessing maximal oxygen intake", JAMA 203(3):201–204: VO2 max = (meters covered − 504.9) ÷ 44.73.

The heart rate method uses Uth N, Sørensen H, Overgaard K, Pedersen PK (2004), "Estimation of VO2max from the ratio between HRmax and HRrest – the Heart Rate Ratio Method", Eur J Appl Physiol 91(1):111–115: VO2 max = 15.3 × maximum heart rate ÷ resting heart rate. It was derived from well-trained men, so estimates for other people are rougher.

All three methods are estimates; individual results vary. VO2 max is shown to one decimal place.

Treadmill calories

The treadmill calculator uses the metabolic equations for walking and running from the American College of Sports Medicine, ACSM's Guidelines for Exercise Testing and Prescription. With speed S in meters per minute and grade G as a fraction (incline percent ÷ 100), walking uses VO2 = 0.1S + 1.8SG + 3.5 and running uses VO2 = 0.2S + 0.9SG + 3.5, in ml of oxygen per kg per minute. The walking equation applies to roughly 1.9 to 3.7 mph and the running equation above 5 mph, or from 3 mph if actually jogging; the calculator uses walking below 4 mph and running from 4 mph unless you choose.

METs are VO2 ÷ 3.5, since 1 MET is 3.5 ml of oxygen per kg per minute. Calories are VO2 × body weight in kg ÷ 1000 × 5 × minutes, using about 5 kcal per litre of oxygen. Both are gross figures that include resting energy use. One pound is exactly 0.45359237 kg.

These are estimates; individual results vary with fitness, and holding the handrails lowers the real cost, so the estimate will be high. Calories are shown to the nearest whole number and METs to one decimal place.

Race predictions

The Riegel column uses Riegel PS (1981), "Athletic records and human endurance", American Scientist 69(3):285–290: T2 = T1 × (D2 ÷ D1)^1.06, where T1 is the time over distance D1 and T2 the predicted time over D2. Riegel fitted it to efforts of 3.5 to 230 minutes, so the race predictor notes when the input time, or a predicted time, falls outside that range. The pace column is the Riegel time divided by the distance.

The VDOT column uses Daniels and Gilbert (1979), as in the VO2 max section: it takes the VDOT of the input result and, for each distance, finds the time that gives the same VDOT.

The marathon time calculator applies the same two formulas to the marathon only. It shows the Riegel and VDOT marathon times as a range, from the faster to the slower, with the pace per mile and per kilometer for each; its splits use the slower of the two predictions at an even pace.

Both assume the runner is equally trained for every distance. These are estimates; individual results vary.

Run/walk intervals

The run walk pace calculator treats the race as repeated cycles of one run interval and one walk interval. The distance covered in one cycle is the run interval ÷ the run pace plus the walk interval ÷ the walk pace, in miles or kilometers. The average pace is the cycle time (run interval + walk interval) ÷ that distance, so it is weighted by distance, not by time. The finish time is the average pace × the race distance, and the number of cycles is the race distance ÷ the distance per cycle.

For 4:00 running at 10:00 per mile and 1:00 walking at 15:00 per mile, one cycle covers 0.4 + 0.0667 miles, so the average is 10:43 per mile; a time-weighted average of the two paces would give 11:00. A walk interval of 0 makes the average equal to the run pace. These are estimates; individual results vary.

Rounding

Calculations keep full precision. Times and paces are rounded to the nearest second only for display, with halves rounded up.