Half Marathon Pace Calculator

Why the half marathon does not close on a whole number in either unit

A half marathon is half of 42.195 km, which is 21.0975 km. That number is awkward in both unit systems at once. In kilometres it is 21 whole kilometres plus 97.5 m. In miles, 21.0975 divided by 1.609344 gives 13.109378728 miles, which is 13 whole miles plus 0.109378728 miles, or 176.028 m. A table that stops at 21 km or at 13 miles therefore stops short of the finish line, and a table that invents a twenty-second kilometre or a fourteenth mile invents distance that the race does not contain. This page keeps the exact distance as the divisor and gives each leftover its own row, so the printed clocks still land on the time you asked for.

The same awkwardness moves the halfway point. Half of 21.0975 km is 10.54875 km, and half of 13.109378728 miles is 6.554689364 miles. The 10 km mark is a familiar checkpoint but it is not halfway: the first 10 km take less than half of the target, and the remaining 11.0975 km take correspondingly more. Treating 10 km as half a race is a small error that compounds across every pace and split printed from it. Here the halfway clock is computed as exactly half of the target, and the two unequal sections around the 10 km mark are printed separately so the difference is visible rather than assumed.

There is a second, quieter kind of leftover: rounding. Race clocks and watches display whole seconds, but dividing a target by 21.0975 almost never produces a whole number of seconds. A watch that shows 299 s/km will not land on the target if you hold it for the full distance, and a watch that shows 481 s/mile will miss it by a different amount, because the two unit systems round at different places. Worse, those two rounded paces are no longer a clean conversion of one another: 299 multiplied by 1.609344 is 481.193856 s/mile, not the 481 the mile column shows. This page prints the exact pace, the whole-second pace, the speed, and each of those gaps, then sets the finish row to the exact target so that no rounding residue is quietly discarded. Nothing on the page asks you to accept a number that cannot be traced back to a printed formula.

Every formula used on this page

Target time in seconds

T = 3600 x hours + 60 x minutes + seconds. The whole page uses this one number; the three time fields feed nothing else.

The distance in kilometres

D_km = 21.0975. Half of the marathon distance of 42.195 km. There is no distance input on this page.

The distance in miles

D_mi = 21.0975 / 1.609344 = 13.109378728... miles. One mile is 1.609344 km exactly. The page keeps the full value in the arithmetic and prints nine decimal places.

The two finish stubs

stub_km = 21.0975 - 21 = 0.0975 km = 97.5 m. stub_mi_m = 21.0975 - 13 x 1.609344 = 0.176028 km = 176.028 m, which is stub_mi_dist = D_mi - 13 = 0.109378728... miles. Both describe the same last section from different whole-number marks.

The halfway point

half_km = 21.0975 / 2 = 10.54875 km; half_mi = D_mi / 2 = 6.554689364 mi. The halfway clock is T / 2 exactly.

Exact pace and speed

p_km = T / 21.0975 seconds per kilometre; p_mi = T / D_mi seconds per mile. speed_kmh = 3600 / p_km; speed_mph = 3600 / p_mi. Pace times speed is 3600 in each unit, so the two pairs check each other in closed form.

The pace a watch shows

w_km = round(p_km) and w_mi = round(p_mi), in whole seconds, printed as minutes and seconds. Both sit next to the exact values so the difference is visible rather than hidden.

Whole-second drift over the full distance

drift_km = w_km x 21.0975 - T; drift_mi = w_mi x D_mi - T. This is how far a constant whole-second pace misses the target. The two drifts differ because the two unit systems round at different places.

The gap between the two rounded paces

cross_gap = w_km x 1.609344 - w_mi seconds per mile; cross_drift = cross_gap x D_mi seconds. This is why converting one watch pace into the other unit does not reproduce the watch pace that was displayed there.

A checkpoint clock and its split

clock(d) = round(T x d / D), where d is the checkpoint distance and D is 21.0975 km or D_mi to match that checkpoint's unit. split(d_i) = clock(d_i) - clock(d_(i-1)), with the first split taken against zero. For the finish row the page uses clock(D) = T exactly, with no rounding, so the splits in a group sum to T.

The sections around the 10 km mark

first_10 = round(T x 10 / 21.0975); last_section = T - first_10. The first covers 10 km, the second covers 11.0975 km, and their sum is T exactly.

Three worked examples

Every row below is recomputed on load by the same function that drives the results above, from the target time printed beside it. Nothing is typed in by hand.

What this page deliberately does not do

• No prediction. You supply the target; this page does not read a recent race or tell you what you can run.
• No other distance and no distance field. 21.0975 km is the one distance on this page.
• No grid of many finish times and no twenty-one-row per-kilometre split table. One target, one set of results.
• No negative splits and no even-versus-negative toggle. Every clock is an even split.
• No printable pace band, no course presets, no aid-station markers and no watch or GPS sync.
• No age or sex grading, no sex selector, and no invented factor table.
• No elevation, hill, weather, heat, humidity or wind correction.
• No training plan, session prescription, weekly volume, coaching or medical advice.
• No good-or-bad label, no cut-point and no grade for any finish time.
• No account, no upload, no saved history and no server-side arithmetic.