Exact arithmetic
Pace, race-distance calculators, the pace board, the pace–speed converter, and even splits are definitions, not models. Pace is finish time divided by distance. Distance is time divided by pace. Time is pace multiplied by distance. Speed is the inverse of pace. Splits divide the target time evenly across each kilometer or mile.
The pace board applies that identity to a list of distances: 5K, 10K, half, marathon, 50K, 100K, 100 miles, and a short track set. Ultra clocks on that page are even-pace times, not ultra predictions. These tools do not estimate fitness, terrain, or weather. If the inputs are correct, the output is the same number you would get with a stopwatch and a measured course.
Rounding
Displayed times round to the nearest second. Displayed speed rounds to one decimal place. Internal arithmetic uses seconds and kilometres so rounding happens once, at display, not at every intermediate step.
Race prediction — Riegel
Equivalent race times use Peter Riegel’s power-law model with exponent 1.06:
T2 = T1 × (D2 ÷ D1)^1.06
T1 and D1 are a recent race time and distance. T2 is the predicted time at distance D2. The formula assumes similar training, similar conditions, and that both distances sit in a range where endurance scales smoothly. It does not account for heat, hills, tactics, or a jump from a 5K to an undertrained marathon. Treat the result as a planning estimate.
Training paces — Jack Daniels VDOT
Training paces come from Jack Daniels’ VDOT system. A recent race distance and time estimate VDOT, an index of current endurance fitness. That value is converted to Easy (E), Marathon (M), Threshold (T), Interval (I), and Repetition (R) paces. Easy covers recovery running and most long runs. Threshold covers tempo work.
The bands are training estimates, not prescriptions. They assume the race you entered is representative. They do not replace coaching, and they do not know your injury history.
Heart rate — Tanaka, %HRmax, and Karvonen
If you do not enter a known maximum heart rate, the calculator estimates it with Tanaka’s formula:
Estimated max HR = 208 − 0.7 × age
Percentage-of-max zones use that maximum. Karvonen zones use heart-rate reserve: (max − rest) × percent + rest. Age-predicted max HR is a population estimate. Individual maximums vary. Heart-rate numbers on this site are training guidance, not medical advice.
Grade-adjusted pace — Minetti 2002
Grade-adjusted pace uses Minetti’s energy-cost polynomial for running versus gradient. Flat-equivalent pace is actual pace multiplied by (flat cost ÷ grade cost). Uphill running costs more energy, so the flat-equivalent pace is faster than the watch pace. Downhill costs less, so the equivalent is slower.
This is an energy-cost estimate, not a GPS measurement and not a prediction of race time on a hilly course.
Age grading — interpolated WMA 2015 factors
“How good is my running time?” compares a result with interpolated World Masters Athletics (WMA 2015) age-grading factors for 5K, 10K, half marathon, and marathon. Age-grade percentage is the statistically elite standard for your age and comparison category divided by your time.
Factors for ages between tabulated points are interpolated. Distances other than those four are not invented. The output is labeled an estimate. It is not an official WMA ranking, not a qualifying standard, and not a statement about talent.
Uneven splits
Negative and positive split tables use two constant paces, switching at halfway. For a drop of 2% (0.02), first-half time is total ÷ (2 − drop) and the second half is the remainder. A positive drop makes the second half faster. Progressive pace interpolates interval pace from the first kilometre or mile to the last, then scales so the finish still matches the goal time.
Kilometre-split and mile-split pages lock the interval. Pace bands print both columns for the official half marathon or marathon distance.
Race equivalents — VDOT versus Riegel
The race time predictor uses Riegel with exponent 1.06. Required race performance is the same formula with distances swapped: the time you need at a known distance to hit a goal. Race equivalents invert Jack Daniels VDOT so other distances match the same fitness index. The two models will not agree exactly. That is expected.
Marathon potential is Riegel locked to 42.195 km. Goal feasibility compares a Riegel prediction with a goal clock and labels the gap close, stretch, or outlier. Those labels are conventions, not a training plan.
Cooper 1968 and 12-minute arithmetic
The Cooper test estimates VO2max from 12-minute distance: (metres − 504.9) / 44.73. Cite Cooper, K. H. (1968), JAMA. The 12-minute run calculator does not estimate VO2; it only converts pace and distance in 720 seconds. VO2 from a race reports Daniels VDOT as an ml·kg⁻¹·min⁻¹ index.
Reverse age grading
Forward age grading reports percent, age-graded time (clock × factor), and the open-class standard. Reverse age grading starts from a target percent and returns the clock time. Running “percentile” on this site is the same WMA band mapping — not a rank in a results database.
Performance overview
The running performance calculator does not add a formula. It takes one recent 5K, 10K, half, or marathon, plus age and comparison category, and shows the same age grade, VDOT, race equivalents, training paces, Riegel marathon potential, and heart-rate zones as the dedicated tools. Optional known max HR replaces Tanaka. Resting HR unlocks Karvonen. Dedicated pages stay for search and for one-question workflows.
Track, intervals, and Yasso 800s
Track tools assume a 400 m lane-1 oval. Lap time is pace per kilometre × 0.4. Interval session time is work × reps plus rest × (reps − 1). Yasso 800s map a marathon goal of H:MM onto 800 m times of MM:SS. That mapping is a named heuristic popularised by Bart Yasso, not a peer-reviewed predictor.
Treadmill belt math
Treadmill pace and speed are the same identity as the pace–speed converter: km/h = 3600 ÷ seconds per kilometre. Distance covered is belt speed × session time. The treadmill pace chart applies that identity to fixed markers (1 km, 1 mile, 5K, 10K, and 20–60 minute sessions). Treadmill intervals convert work speed and work duration into distance, then reuse the track interval session clock.
Treadmill incline uses Minetti 2002 grade-adjusted pace with belt pace and deck percent. Jones and Doust (1996) reported that about 1% treadmill grade can better match outdoor level-ground energetics for some runners; that note is guidance in the copy, not a second formula on this site.
Cadence, stride, and steps
Running cadence here is steps per minute counting both feet. Speed in metres per second equals (cadence ÷ 60) × stride length in metres. Steps per kilometre are 1000 ÷ stride. Total steps are distance in metres ÷ stride. There is no universal 180 spm target and no height-based stride guess.
Hills and elevation
Segment grade-adjusted pace turns a watch pace on a grade into flat-equivalent pace with Minetti 2002: flat pace = hill pace × (flat cost ÷ grade cost). Hill pace is the inverse: the pace to run on a grade to match a flat target effort. The hills board applies that conversion across common distances and a fixed grade sweep, and reports elevation gain as distance × grade.
Elevation gain and gain per kilometre are exact arithmetic on the metres and distance you type. Vertical speed and VAM are gain ÷ time (m/h and ft/h). Hill repeats convert flat target pace to hill pace, then reuse the interval session clock.
Elevation-adjusted finish and the uphill race predictor form an average grade from net gain ÷ distance, then multiply a flat clock by Minetti cost(grade) ÷ cost(0). When the predictor starts from a different race distance, Riegel scales first. Trail versus road pace multiplies road pace by a factor you enter. None of these pages fetch GPS elevation. Net gain ignores equal ups and downs; heat and pacing strategy are not modelled.
The flat sea-level marathon equivalent calculator inverts that hill multiplier on the raced distance, divides out the altitude pace factor, then converts the flat sea-level race time into a marathon with Daniels VDOT and Riegel shown side by side. Compare mode runs the same pipeline on two races. It is a progress-tracking estimate, not a GPS course model.
Training volume
Weekly, monthly, and annual volume tools are exact arithmetic. A month is four weeks and a year is fifty-two weeks unless a page says otherwise. Mileage increase multiplies current volume by (1 + percent ÷ 100). The 10% rule is the same identity with percent fixed at 10 — a popular heuristic, not injury science. Long-run percentage is long ÷ weekly. Polarized 80/20 splits a weekly total into easy and hard shares you classify. Time on feet sums session durations. Frequency maps runs per week and average distance to weekly volume.
Energy and fueling
Running calories use the ACSM running VO2 equation: 0.2 × speed (m/min) + 0.9 × speed × grade fraction + 3.5, then convert ml O₂ to kcal with body mass (about 5 kcal per litre of oxygen). Race energy pages lock half-marathon or marathon distance on the same engine. Carbohydrate targets use ISSN/ACSM-style duration bands (~30–60 g/h for roughly 1–2.5 hours, up to ~90 g/h for longer events). Gel count divides total carbs by grams per gel. Fuel timelines place even takes from a first-gel clock. Caffeine timing reports 3–6 mg/kg and a typical 30–60 minute pre-event window. Every energy result is an estimate, not medical advice, and not a personalised nutrition plan.
Environmental conditions
Conditions tools multiply a cool-weather goal pace by published simplification factors. Heat uses a capped slowdown versus a cool reference temperature (Ely-style). Humidity adds a modifier when it is warm and humid. Wind uses a linear percent per m/s (positive headwind, negative tailwind). Altitude applies roughly 1% slower per 300 m above about 600 m. The conditions board multiplies heat, humidity, wind, and altitude together. These are planning estimates, not race-day physiology, and they are not grade-adjusted pace.
Weather API
The race weather calculator can optionally fetch an Open-Meteo hourly forecast for temperature, relative humidity, wind speed, and precipitation. The forecast is input only. Pace adjustment still runs in the browser with the same conditions engine. If the request fails, typed values continue to work. There is no map product and no paid weather vendor on this page.
Fun tools
Fun calculators reuse deterministic engines: marathon steps from stride and 42.195 km, distance in one hour from pace, marathon and famous-distance equivalents, playlist and song counts versus run duration, documented elite references (not a live results database), and lifetime distance from weekly volume × years. Tone can be lighter; citations and arithmetic stay honest.
What results are not
Nothing on this site is medical advice, nutritional advice, a coaching plan, or an official ranking. Exact-arithmetic tools answer a definition. Model-based tools answer a published formula with the inputs you typed. Use them for training guidance, then apply judgement.
Browse the calculators or read about the lab.
Citations
- Riegel, P. S. (1977). Time predicting. Runner's World. See also Riegel, P. S. (1981). Athletic records and human endurance. American Scientist, 69(3), 285–290.
- Daniels, J. Daniels' Running Formula. Human Kinetics.
- Tanaka, H., Monahan, K. D., & Seals, D. R. (2001). Age-predicted maximal heart rate revisited. Journal of the American College of Cardiology, 37(1), 153–156.
- Karvonen, M. J., Kentala, E., & Mustala, O. (1957). The effects of training on heart rate. Annales Medicinae Experimentalis et Biologiae Fenniae, 35(3), 307–315.
- Minetti, A. E., Moia, C., Roi, G. S., Susta, D., & Ferretti, G. (2002). Energy cost of walking and running at extreme uphill and downhill slopes. Journal of Applied Physiology, 93(3), 1039–1046.
- World Masters Athletics. Age-grading factors, 2015 tables. This site uses an interpolated subset for 5K, 10K, half marathon, and marathon only.
- Cooper, K. H. (1968). A means of assessing maximal oxygen intake. JAMA, 203(3), 201–204.
- Yasso 800s as popularised by Bart Yasso / Runner's World. A workout heuristic, not a marathon prediction model.
- Jones, A. M., & Doust, J. H. (1996). A 1% treadmill grade most accurately reflects the energetic cost of outdoor running. Journal of Sports Sciences, 14(4), 321–327.
- American College of Sports Medicine. Running metabolic equations (VO2 from speed and grade) as used in ACSM guidelines for exercise testing and prescription.
- International Society of Sports Nutrition position stands on nutrient timing and caffeine — carbohydrate g/h bands and 3–6 mg/kg caffeine ranges cited as planning estimates, not prescriptions.
- Ely, M. R., Cheuvront, S. N., Roberts, W. O., & Montain, S. J. (2007). Impact of weather on marathon-running performance. Medicine & Science in Sports & Exercise, 39(3), 487–493. Heat factor here is a simplified planning estimate.
- Open-Meteo forecast API for optional race-weather inputs (temperature, humidity, wind, precipitation). Pace math remains local.