Methodology

This page exists so that no calculation on calculaterun is a black box. Each calculator also carries its own formula and worked example; this is the consolidated reference.

Race time prediction — Riegel

Peter Riegel published this relationship in 1977 after observing that a runner’s time-versus-distance curve is close to a straight line on logarithmic axes. The exponent 1.06 means each doubling of distance costs roughly four percent of pace.

Reliability. Good between 3 km and 30 km for a runner trained across that range. Degrades sharply when the target distance is more than about three times the input distance, and consistently over-predicts marathon performance for runners without substantial long-run volume, because the model contains no term for glycogen depletion.

Training paces — Daniels and Gilbert VDOT

Published by Jack Daniels and Jimmy Gilbert in Oxygen Power (1979). The first equation gives the oxygen demand of a running velocity; the second gives the fraction of maximum sustainable for a given duration. Training zones are then generated by solving the first equation backwards at fixed percentages of VDOT: easy 62–72%, marathon 79–84%, threshold 86–88%, interval 97–100%, repetition 105–110%.

Reliability. Built from data on trained runners and best applied to them. Assumes flat terrain and moderate conditions. VDOT is a performance index, not a laboratory VO₂max, and will differ from a lab value because it embeds running economy.

Metabolic cost of grade — ACSM

The American College of Sports Medicine running equation. The vertical coefficient is 4.5 times the horizontal, which is why modest inclines change effort so much.

Reliability. Validated for running above approximately 134 m/min on grades up to about 15%. Substantially underestimates the muscular cost of downhill running, where eccentric loading rather than oxygen consumption is the limiting factor.

Heart rate zones — Karvonen, Tanaka, threshold anchoring

Where a measured lactate threshold heart rate is available, zones are anchored to it directly as percentages of threshold, which is the most physiologically meaningful of the three approaches offered.

Reliability. Age-based maximum estimates carry a standard deviation of roughly 10–12 bpm and should be treated as a last resort. Karvonen improves on raw percentage-of-maximum by accounting for resting heart rate. Threshold anchoring is preferred because the anchor is measurable in the field and responds to training.

Sweat rate

Relies on the near-equivalence of one litre of sweat to one kilogram of mass. Assumes no urination during the measurement period and dry clothing at both weigh-ins.

Reliability. Accurate for the specific conditions of the test only. Sweat rate varies with temperature, humidity, intensity and acclimatisation state, so a rate measured in spring will underestimate summer losses.

Unit conventions

One mile is taken as exactly 1.609344 km. The marathon is 42.195 km and the half marathon 21.0975 km, rather than the rounded 26.2 and 13.1 miles used on race merchandise.

Corrections

If you believe a formula is implemented incorrectly, get in touch with your inputs and expected output. Errors are corrected and noted rather than silently amended, as set out in the editorial policy.