Training plans and race-day pacing

Methodology: formulas and plan rules

Last updated 2026-10-20

Every number on RunPacePlan comes from code that runs in your browser. This page lists what that code does: the formula each calculator uses, where it comes from, how far to trust it, and the rules the training plan generator follows. The same code is tested against answers worked out independently (worked examples from other publications, hand calculations and measured values from the original papers).

Pace, time and distance

Pace is time divided by distance. We use the international mile (1,609.344 m), the marathon distance of 42.195 km and a half marathon of exactly half that (21.0975 km). Speed in mph or km/h is converted to pace as 60 ÷ speed minutes per mile or km. Split tables divide the goal time in proportion to distance, so even splits always add up to the goal exactly. For negative or positive splits the faster half is a chosen percentage quicker than the slower half, and the two halves still add up to the goal time.

Race time prediction

Riegel (1981). Peter Riegel’s formula, published in American Scientist 69(3): 285–290, predicts a time T2 over distance D2 from a time T1 over D1: T2 = T1 × (D2 ÷ D1)^1.06. It works best between about 1,500 m and the half marathon. For the marathon it is often optimistic, especially for runners with low weekly mileage, and it was not built for efforts much longer than about four hours.

Daniels & Gilbert (1979). Jack Daniels and Jimmy Gilbert published two equations in Oxygen Power: Performance Tables for Distance Runners:

  • the oxygen cost of running at v metres per minute: VO2 = −4.60 + 0.182258 v + 0.000104 v²
  • the fraction of maximum oxygen uptake that can be held for t minutes: F = 0.8 + 0.1894393 e^(−0.012778 t) + 0.2989558 e^(−0.1932605 t)

Dividing the first by the second gives an estimate of VO2 max from a race, which we call the Race Fitness Score. A 10K in 40:00, for example, gives 51.9. To predict another distance we search for the time that gives the same score. We implement the published equations ourselves; we do not use or reproduce the VDOT tables, and “VDOT” (a trademark of The Run SMART Project) is not used as a name on this site.

Training paces

From the Race Fitness Score we define six training paces. The definitions are our own:

  • Easy: 15–30 % slower per kilometre than your marathon-equivalent pace.
  • Long run: 10–25 % slower than marathon-equivalent pace.
  • Marathon pace: the marathon time that matches your score, as a pace.
  • Threshold: the pace you could race for about 60 minutes.
  • Interval: the pace you could race for about 10 minutes (close to 3K race pace).
  • Repetition: your score-equivalent mile race pace.

Treadmill and incline

Incline. Minetti and colleagues measured the energy cost of running on slopes from −45 % to +45 % (Journal of Applied Physiology 93: 1039–1046, 2002) and fitted the cost per metre at gradient i (as a fraction) as C(i) = 155.4 i^5 − 30.4 i^4 − 43.3 i^3 + 46.3 i^2 + 19.5 i + 3.6 J/kg/m. At the same energy use per minute, speed × C stays the same, so the flat-ground speed that matches speed v at incline i is v × C(i) ÷ C(0). The results are averages from a group of runners and are labelled as estimates.

1 % incline. Jones and Doust (Journal of Sports Sciences 14: 321–327, 1996) found that a 1 % treadmill grade best matched the energy cost of running outdoors, for speeds from 2.92 to 5.0 m/s (about 6.5 to 11.2 mph). Treadmill speed itself is converted to pace with plain arithmetic.

Heart rate

  • 220 − age is the traditional rule of thumb.
  • Tanaka: 208 − 0.7 × age (Tanaka, Monahan & Seals, Journal of the American College of Cardiology 37: 153–156, 2001).
  • Gulati: 206 − 0.88 × age, from a study of women (Gulati et al., Circulation 122: 130–137, 2010).

Individual maximum heart rates scatter widely around any formula, typically by about 10 beats per minute or more, so a measured maximum is always better. Zones by percentage of maximum use 50–60–70–80–90–100 %. The Karvonen method (Karvonen, Kentala & Mustala, 1957) applies the same percentages to the heart rate reserve: target = resting + percentage × (maximum − resting). Zone 2 is 60–70 % in both methods.

Not medical advice. If you have heart disease, chest pain, dizziness, are pregnant or take medication that affects heart rate, ask a doctor before training by heart rate.

VO2 max and cadence

VO2 max from a race uses the Daniels & Gilbert equations above. The Cooper test (Cooper, JAMA 203: 201–204, 1968) estimates VO2 max from the distance run in 12 minutes: (metres − 504.9) ÷ 44.73. We do not publish VO2 max ratings by age and sex because we have not found a reference table whose licence allows it; the chart on the VO2 max page shows race times computed with the same equations.

Cadence is steps per minute. Heiderscheit and colleagues (Medicine & Science in Sports & Exercise 43: 296–302, 2011) found that raising step rate 5–10 % at the same speed reduced loading at the knee and hip, which is why the cadence tool shows +5 % and +10 % targets.

Calories burned running

The running calories calculator uses the ACSM metabolic equation for running (ACSM’s Guidelines for Exercise Testing and Prescription): oxygen use in ml per kg per minute = 0.2 × speed (m/min) + 0.9 × speed × incline (as a fraction) + 3.5. Energy is taken as about 5 kilocalories per litre of oxygen. The total is everything burned during the run; the figure above resting subtracts the 3.5 resting term. The equation is meant for running speeds above about 134 m/min (8 km/h), and it is an average, so your own result will differ.

Boston qualifying times

The qualifying standards, cut-offs and age rule come from the Boston Athletic Association’s qualifying page (baa.org), read on 20 October 2026, for the race on 19 April 2027. We check them each September when the next race’s standards are published.

Training plan rules

Plans are generated from the rules below; no published training plan was used as input. Change the number of weeks, days or level and the whole plan changes with them. The numbers are our own settings, chosen to be conservative.

  1. Starting and peak weekly distance depend on the race and level (for example a 16-week intermediate marathon plan on 5 days starts near 33 km and peaks near 71 km a week), scaled down for fewer running days, and never below what the running days need.
  2. No week is more than 10 % above the largest week before it. The 10 % figure is a common rule of thumb rather than a proven injury threshold: a randomised trial in novice runners (Buist et al., American Journal of Sports Medicine, 2008) did not find fewer injuries with a slower build, so we treat it as a ceiling, not a target.
  3. Every fourth week is a cutback week at about 80 % of the week before, unless it falls just before the taper.
  4. The long run is at most 30–45 % of the week for 5K to half marathon plans (30 % on 6 days, 45 % on 3 days) and 40–50 % for marathon and 50K plans, and never longer than a set maximum: for example 32 km (20 miles) for most marathon plans and 35 km for advanced ones.
  5. Taper: one week for 5K and 10K, two for the half marathon, three for the marathon and 50K, each smaller than the last; race week ends with the race.
  6. Hard sessions (hills, tempo, intervals, race pace): none to two a week depending on level and phase, never the day before or after the long run, and capped at 30 % of the week.
  7. Phases: base (first 40 % of the build: easy running, strides, hills), build (next 40 %: tempo and intervals), race-specific (last 20 %: race-pace work, and marathon-pace segments in long runs), then taper.
  8. Paces come from your goal time through the training paces above. Without a goal the plan describes each run by effort.
  9. Easy runs are at least 3 km, and no run is longer than the week’s long run.

The beginner run-walk plan grows the running interval from 1 minute towards 30 minutes of continuous running over 8–10 weeks, with walking breaks shrinking from 2 minutes to none. Running time per session never goes down from one week to the next.

Automated tests check every combination the plan pages offer (race, level, 3–6 days, every plan length, with and without a goal time) against rules 2–9.