Running Power Calculator

What this page does

A running power calculator answers a different question from a calorie count: not how much energy a run costs in total, but the rate at which the body spends it, per kilogram, while you run. Type a pace, a gradient and a body mass and the page returns watts per kilogram and the absolute watts, on the energy-cost polynomial from Minetti et al. (2002) that this site already uses on its gradient and calorie pages. The direction also runs backwards: type a power per kilogram and the page returns the flat pace it corresponds to.

What it is not is a watch readout. A watch or a foot pod reports a mechanical number derived from its own accelerometers and its own undisclosed model; this page reports metabolic power, which is the energy the body spends per second per kilogram of body mass. The two are not the same quantity and this page does not convert between them, because the conversion needs an efficiency factor it has no source for. The running calorie calculator prices the total cost of a run in kilocalories; the treadmill pace calculator prices a belt incline into an outdoor pace; the marathon split calculator prices a whole elevation profile into splits. This page is the rate, in watts per kilogram, and the way back from a rate to a pace.

Everything runs in your browser. Nothing you type is sent to a server, there is no account and no upload, and the numbers never leave the page. The arithmetic is one published polynomial and three unit definitions, all written up below with the citations behind them.

How the numbers are calculated

One published fit carries the whole page. Minetti, Moia, Roi, Susta and Ferretti (2002) measured the energy cost of running on a treadmill at gradients from −45% to +45% and published it as a polynomial in the gradient i, in joules per kilogram per metre travelled:

C(i) = 155.4i⁵ − 30.4i⁴ − 43.3i³ + 46.3i² + 19.5i + 3.6

The cost is per metre, so the rate of energy expenditure is the cost times the speed. With m the body mass in kilograms and v the speed in metres per second:

P/kg = C(i) × v    P = C(i) × m × v

Both outputs on this page come from those two lines. Watts per kilogram is the first; the absolute watts are the first multiplied by the body mass. Nothing else is applied — no efficiency factor, no correction for the shoe, the surface or the wind.

The flat case is an identity worth knowing

At zero gradient the polynomial returns exactly 3.6 J/kg/m, and 3.6 km/h is exactly 1 m/s. So on the flat, watts per kilogram and speed in km/h are the same number: P/kg = 3.6 × (km/h ÷ 3.6) = km/h. A runner at 5:00 per kilometre is at 12.0 km/h and therefore at 12.0 W/kg; a runner at 6:00 per kilometre is at 10.0 km/h and 10.0 W/kg. That identity is exact and it is the quickest way to check any number on this page.

A worked example: 5:00 per kilometre, flat, 70 kg

5:00 per kilometre is 12.0 km/h, which is 3.3333 m/s. On the flat C(0) = 3.6, so P/kg = 3.6 × 3.3333 = 12.0 W/kg, and P = 12.0 × 70 = 840 W. Put the same pace on a +5% gradient and C(0.05) = 4.685196, so P/kg = 4.685196 × 3.3333 = 15.6 W/kg and the absolute power is 1,093 W — a 30% increase for the same speed, because the fit says a +5% gradient costs 1.30 times what the flat costs.

The reverse is the flat identity read backwards: 12.0 W/kg is 12.0 km/h, which is 5:00 per kilometre and 8:03 per mile. No iteration is needed and none is done.

The cross-check that ties this page to the calorie page

At 10.0 km/h on the flat, a 70 kg runner is at 10.0 W/kg, which is 700 W. Hold that for one hour and the energy is 700 × 3600 = 2,520,000 J, which at 4,184 J per kilocalorie is 602.3 kcal — the same 602 kcal the running calorie calculator prints for 10 km at 10 km/h for a 70 kg runner. Two pages, one fit, and the numbers meet. That is deliberate: this page does not carry a second model.

• i — the gradient as a fraction, so +5% is 0.05. Accepted from −0.45 to +0.45.
• v — speed in metres per second. 1 km/h = 1/3.6 m/s; 1 mph = 1609.344 ÷ 3600 = 0.44704 m/s.
• C(i) — the energy cost, in joules per kilogram per metre travelled. 3.6 at zero gradient.
• P/kg — metabolic power per kilogram of body mass, in watts per kilogram.
• P — the same rate for the whole body, in watts.

One honesty note on the fit, the same one the other gradient pages on this site carry. The paper reports a level cost of about 3.40 J/kg/m while the published polynomial returns 3.6 at zero gradient, so the polynomial runs about 6% high on the flat. This page uses the polynomial as published rather than rescaling it, because a rescaled version would be a number nobody published.

The gradient table, and why a descent is not free

The table above prices eight gradients at the pace you typed. The third column is the only one that needs no explanation: it is C(i) ÷ C(0), how many times the flat cost that gradient is. Uphill it climbs fast — +2% is 1.11 times, +5% is 1.30 times, +10% is 1.66 times — and the reason is that lifting the body is a cost the flat does not have at all, so a small percentage of climb is a large percentage of the total.

Downhill is the interesting half. The cost falls to about 0.76 times the flat at −5% and about 0.60 times at −10%, but it does not keep falling: the polynomial bottoms out somewhere around −18% to −20% at about half the flat cost, and then climbs again, because running down a very steep slope costs braking and stabilising work that a gentle descent does not. A 20% descent is not 20% cheaper than the flat; it is about half. The table above stops at −10% for readability, but the input accepts anything down to −45%, and the curve does turn back up.

Two limits worth stating plainly. The fit is a treadmill fit on ten runners, so it is a fit and not a law. And the gradient is one constant number here — a real course changes gradient every few hundred metres, and pricing a whole profile kilometre by kilometre is what the marathon split calculator and the race time calculator do.

Body mass and absolute power

Watts per kilogram is the number that travels: it is the same for two runners at the same pace on the same gradient whatever they weigh, because the fit is per kilogram by construction. Absolute watts do not travel. At 12.0 W/kg a 50 kg runner is at 600 W and a 110 kg runner is at 1,320 W, and both are running 5:00 per kilometre on the flat. The second table above prints that line for every 5 kg from 50 to 110 at the power you typed.

This is also why a heavier runner's watch shows a bigger number for the same pace and why comparing absolute watts between two people says almost nothing. The per-kilogram figure is the comparable one, and even then only against the same gradient.

The reverse direction: power back to a pace

The reverse block takes a metabolic power — in watts per kilogram, or in watts with the body mass from the form above — and returns the flat pace. On the flat the arithmetic is the identity from earlier, so 14.0 W/kg is 14.0 km/h, which is 4:17 per kilometre and 6:54 per mile. Watts are divided by the body mass first: 980 W for a 70 kg runner is 14.0 W/kg, and the same pace follows.

This direction is deliberately flat only. A gradient does not change the pace a given power buys in any way this page can price, because pricing it means turning a gradient into a pace — and that is exactly the job of the treadmill pace calculator for a belt incline, the treadmill pace conversion for a flat belt, and the marathon split calculator for a road profile. Here a gradient only changes what a pace costs, which is the forward direction.

Supported range and limits

• Pace: 2:30 to 10:00 per kilometre, which is 6.0 to 24.0 km/h. Per mile that is 4:01 to 16:06. Outside the range the page refuses rather than prints, because below 6.0 km/h you are walking and the fit on this page is the running fit.
• Gradient: −45% to +45%, the range the polynomial was fitted over. Values outside are refused, not clamped silently.
• Body mass: 40 to 150 kg.
• Reverse power: 6.0 to 22.0 W/kg on the flat, which is 6.0 to 22.0 km/h. In watts the same limits apply after dividing by the body mass.
• Rounding: watts per kilogram and watts to one decimal, paces to the nearest second, speeds to two decimals. One rounding, at the end.
• Cost: none. No account, no email, no paywall, no daily limit, no request to a server.
• Privacy: every number is computed in your browser. Nothing you type is sent anywhere or stored — reload the page and it is gone.
• Works offline once the page has loaded, and on a phone.

What this page deliberately does not do

• No watch or foot-pod comparison. A device reports mechanical power from its own model; this page reports metabolic power from one published fit. Converting one into the other needs an efficiency factor, this page has no source for one it can stand behind, and rather than ship a made-up ratio it prints the quantity it can cite and says which one that is.
• No pace from a gradient. The gradient changes what a pace costs, in watts, and that is the forward direction. Turning a gradient into a pace is the treadmill pace calculator for a belt incline, the treadmill pace conversion for a flat belt, and the marathon split calculator for a road profile.
• No walking. The fit is the running fit. Speeds below 6.0 km/h are refused rather than priced with it; the same paper has a separate walking fit, which this page does not carry.
• No kilocalories. The total cost of a run in kcal is the running calorie calculator. This page prints a rate, in watts, and nothing about fat, fuel split or weight loss.
• No weather correction. Heat, headwind and humidity change the cost of a run. This page prices none of them, and adding an unsourced adjustment would make the number look more precise than it is.
• No heart-rate model. Heart rate is not an input and is not estimated from the answer. Two runners with the same mass at the same pace on the same gradient get the same number here whatever their heart rates did.
• No training plan and no verdict. No sessions, no intervals, no durations, no weekly volume, no zones, and no statement that a power is good, bad, high or low. A watt is arithmetic, not a grade.
• No prediction. This page converts a pace into a rate and back. What you are capable of over a race distance is the race time predictor on the home page.

Questions