Methodology & Sources
Every number on this site comes from a formula printed here, worked from a source named here. No hidden adjustments, no borrowed output from another calculator — if you want to check a result with a pencil, this page gives you everything you need.
Last reviewed: 10 November 2026
Start with speed and pace
Speed and pace are the same fact read two ways. Speed says how much ground you cover in an hour; pace says how long one unit of that ground takes you.
The speed can be in miles per hour for a pace per mile, or in kilometres per hour for a pace per kilometre — the division is identical, only the unit changes. Run the sum backwards and pace becomes speed again: speed = 3,600 ÷ pace in seconds.
Worked check: a belt at 6.5 mph gives 3,600 ÷ 6.5 = 553.8 seconds per mile, which this site rounds and displays as 9:14 per mile. The same belt speed is 10.46 km/h, and 3,600 ÷ 10.46 = 344 seconds, or 5:44 per kilometre. Both answers describe one run; neither is more correct than the other.
Units, converted exactly
Unit conversion on this site is not estimated and never rounded at the definition stage.
One mile is 1.609344 kilometres exactly, by definition — the figure set out in NIST Special Publication 811, built from the exact yard of 0.9144 metres and the 1,760 yards in a mile. To turn mph into km/h, multiply by 1.609344. To turn km/h into mph, divide by the same number. A pace per mile becomes a pace per kilometre by dividing by 1.609344, because the kilometre is the shorter unit and the same speed covers it sooner.
That single constant sits underneath every conversion on the site — the converter, the chart, the finish times and the walking table all call the same code, so two pages can never disagree about what 7 mph means.
Grade-adjusted pace
Climbing costs energy, so a pace run uphill on the belt is a harder effort than the same pace on the flat. Grade-adjusted pace (GAP) answers the fair comparison: what flat pace would this incline effort equal?
Cr(i) = 155.4i5 − 30.4i4 − 43.3i3 + 46.3i2 + 19.5i + 3.6 J/kg/m
Flat-equivalent speed = belt speed × Cr(i) ÷ Cr(0)
Grade-adjusted pace = belt pace ÷ (Cr(i) ÷ Cr(0))
The polynomial is the running curve published by Minetti and colleagues in 2002 (Journal of Applied Physiology 93(3):1039–1046). It gives the energy cost of running, Cr, in joules per kilogram per metre, as a function of the gradient i — rise divided by run, so a 10% grade is i = 0.10. The curve was fitted over gradients from −45% to +45%, far wider than any treadmill range, and on the flat it returns Cr(0) = 3.6 J/kg/m. Dividing the cost at your grade by the flat cost gives a pure effort ratio, and that ratio is what scales speed up and pace down.
| Grade | Energy cost (J/kg/m) | vs flat |
|---|---|---|
| 1% | 3.80 | +5.5% |
| 5% | 4.69 | +30.1% |
| 10% | 5.97 | +65.8% |
Check values from the polynomial above: Cr(0.01) = 3.80, Cr(0.05) = 4.69, Cr(0.10) = 5.97, each against the flat cost of 3.6.
Two honest notes. First, Minetti and colleagues published an energy-cost curve, not a pace table. Turning cost into an equivalent flat pace — the construction set out above — is the standard grade-adjusted pace construction used across running tools, and it is this site's step, stated openly rather than attributed to the paper. Second, the paper also fits a separate curve for walking. This site uses the running curve for grade-adjusted pace at every belt speed and says so, instead of quietly switching models at slower speeds where nobody would notice.
The 1% rule
The best-known finding in treadmill research — and the one most often repeated with more certainty than the study itself carries.
The source is Jones and Doust (1996), Journal of Sports Sciences 14(4):321–327 (PMID 8887211). Nine trained male runners ran six-minute bouts at speeds from 2.92 to 5.0 metres per second — roughly 6.5 to 11.2 mph, or paces from about 9:11 to 5:22 per mile — on treadmill grades of 0%, 1%, 2% and 3%, and on a level outdoor road, with oxygen use measured throughout.
At a 1% grade, oxygen use on the treadmill was not significantly different from road running at any speed tested. That is the whole finding, and it is why the outdoor converter on this site uses 1% as its default bridge between belt and road. The study also found that at the two slowest speeds tested, a 0% grade was not significantly different from the road either — the paper does not say slower running needs no correction, only that at those speeds both settings sat inside the same statistical noise.
The limits matter as much as the headline. The runners were nine trained men. The slowest speed tested was still a genuine run, so walking and easy jogging were never tested and the rule says nothing direct about them. And despite how often it is credited elsewhere, the 1% rule is not an ACSM position — it is one well-run laboratory study, used here as exactly that.
Heart-rate zones
Heart-rate zones start from a maximum heart rate nobody has measured. This site shows both standard estimates and labels both as estimates.
The older estimate is the Fox formula, HRmax = 220 − age, dating to 1971. It survives because it is easy to remember, not because it is precise: it began as a rule of thumb, and individuals scatter around it by roughly ±10–12 beats per minute. The newer estimate is Tanaka and colleagues (2001): HRmax = 208 − 0.7 × age, drawn from a meta-analysis of 351 studies covering 18,712 subjects, with a standard error of estimate of about 7 beats per minute. The zone calculator shows the Tanaka figure alongside the Fox figure so the gap between them is visible instead of hidden.
Whichever maximum is used, the bands applied to it are the intensity classes of the American College of Sports Medicine (Garber et al., 2011):
| Zone | Class | % of HRmax |
|---|---|---|
| Zone 1 | Very light | Under 57% |
| Zone 2 | Light | 57–63% |
| Zone 3 | Moderate | 64–76% |
| Zone 4 | Vigorous | 77–95% |
| Zone 5 | Near maximal | 96% and above |
Because the maximum itself is estimated, every zone boundary inherited from it is an estimate too. A measured maximum from a supervised test outranks both formulas.
Pace zones
Pace zones are built from a threshold pace — the fastest pace you can hold in a controlled, sustainable effort — expressed as percentages of threshold speed.
| Zone | % of threshold speed |
|---|---|
| Zone 1 — Recovery | 65–79% |
| Zone 2 — Endurance | 80–87% |
| Zone 3 — Tempo | 88–92% |
| Zone 4 — Threshold | 93–100% |
| Zone 5 — VO2 max | 101–106% |
Labelled plainly: these bands are a coaching convention, not a governing standard. No federation or standards body publishes official pace zones, and different coaches draw the lines in slightly different places. The percentages above are the convention this site uses, printed here so you can see exactly what the zone calculator is doing. Because the bands are percentages of speed, the paces run the other way: Zone 2 at 80–87% of threshold speed means a pace slower than threshold pace, found by dividing threshold pace by 0.87 and by 0.80.
Distance, time and steps
The remaining tools are the same identity rearranged, plus one estimate that is labelled every time it appears.
Hold 6 mph for 30 minutes and you cover 6 × 0.5 = 3 miles. Turned around, time = distance ÷ speed: 3 miles at 6 mph takes half an hour, however the question is phrased. Race finish times on this site are this sum applied to the standard distances — 5 km, 10 km, the half marathon at 21.0975 km and the marathon at 42.195 km.
Steps are different: they are an estimate. The distance calculator offers steps = distance ÷ stride length. Stride varies with height, speed and fatigue, so if you do not enter your own, the calculator defaults to a stride of 0.75 metres and labels the result an estimate on the page. At that default, a 5 km run reads as roughly 6,667 steps — a planning figure, not a count, and your watch, which actually felt each stride, outranks it.
Rounding and checking
Small decisions, made the same way on every page.
Paces are rounded to the nearest second for display — 553.8 seconds becomes 9:14, never 9:13 by truncation and never a decimal second a runner cannot use. Speeds are shown to one decimal place, matching the steps most treadmill consoles actually offer.
Before any version of this site is published, every worked example printed on its pages is reproduced through the site's own JavaScript in a test harness — the same code the calculators run in your browser, fed the printed inputs and checked against the printed answers. A formula change that broke an example would fail that check before the page shipped. When a reader finds a number that still slipped through, the correction is fixed in the code and logged openly with its date. Found one yourself? Report an error — reader corrections are part of the method.
Sources
Every source the calculations above rest on, named and linked. The name is the link — open it and check the same document.
- Jones & Doust (1996), Journal of Sports Sciences
“A 1% treadmill grade most accurately reflects the energetic cost of outdoor running” — nine trained runners, treadmill grades 0–3% against a level road, speeds 2.92–5.0 m/s. The source of the 1% rule.
- Minetti et al. (2002), Journal of Applied Physiology
“Energy cost of walking and running at extreme uphill and downhill slopes” — the fifth-order polynomial for the energy cost of running at gradients from −45% to +45% that the grade-adjusted pace calculations use.
- Tanaka, Monahan & Seals (2001), Journal of the American College of Cardiology
“Age-predicted maximal heart rate revisited” — a meta-analysis of 351 studies (18,712 people) behind the HRmax = 208 − 0.7 × age formula, shown next to the traditional 220 − age.
- American College of Sports Medicine — Guidelines for Exercise Testing and Prescription
The intensity classification (Garber et al., 2011) behind the heart-rate zone bands: very light under 57% of HRmax, light 57–63%, moderate 64–76%, vigorous 77–95%, near maximal 96% and above.
- American Heart Association
Walking guidance, including brisk walking at about 3 mph as the everyday moderate-intensity benchmark used on the walking page.
- National Institute of Standards and Technology
The exact unit definitions: 1 mile = 1,609.344 metres (SP 811), from which every mph ↔ km/h conversion on this site follows.
Formulas checked against Jones & Doust (1996) and Minetti et al. (2002) — every source is named on the Methodology page. Found a wrong number? Report an error.