Half Marathon Distance

Why the three rounded descriptions do not land on the definition

The standard is a metric number: 21097.5 m, exactly half of 42195 m. The three descriptions in common use were not built from that number, so none of them lands on it. One mile is exactly 1609.344 m and one yard exactly 0.9144 m. Thirteen point one miles is therefore 13.1 x 1609.344 = 21082.4064 m, which is 15.0936 m shorter than the defined distance. That is a little over fifteen metres, roughly the length of a city bus, and it is the largest of the three gaps.

The metric rounding goes the other way. Twenty-one point one kilometres is 21100 m, which is 2.5 m longer than 21097.5 m. Both numbers are descriptions of the same race, and they are wrong in opposite directions: one is short and one is long. This page states each gap in metres with an explicit direction, so the difference between a description and a definition is visible instead of being absorbed into a decimal point.

The third description is the closest one, and the most interesting. Thirteen miles are 20921.472 m, leaving 176.028 m before the finish. That leftover divided by the exact 0.9144 m in a yard is 192.506561679788 yd, so the honest imperial phrasing is 13 miles 192.506561679788 yards. Rounding that to 192.5 yards gives 13 x 1609.344 + 192.5 x 0.9144 = 21097.494 m, which is 0.006 m, or 6 mm, short of 21097.5 m. Six millimetres is far too small to matter on a measured road course, but it is a real difference, and it is the reason this page keeps the metric constant as the only standard. None of the three rounded phrases is used as an input anywhere in the arithmetic.

The constant also splits cleanly in whole pieces. Twenty-one whole kilometres leave 97.5 m. Thirteen whole miles leave 176.028 m. Fifty-two whole laps of 400 m leave 297.5 m, because 52 x 400 is 20800 m. These are three descriptions of the same short final section measured from different whole-number marks, not three different finish lines. Exact yards and feet are printed above as well: 21097.5 / 0.9144 = 23072.506561679791 yd and 21097.5 / 0.3048 = 69217.519685039369 ft. Every one of those is a division of the one constant, not a value borrowed from a table.

What you can change is only the number of equal parts. Dividing 21097.5 m by a part count gives a length in metres, and that is the canonical number the tool stores; the display unit reformats it for reading and never writes back into the arithmetic. Because each part is exactly 42195 / (2n) metres, n parts sum back to 42195 / 2 = 21097.5 m by algebra, with no rounding inside the chain. If the parts are cut to whole metres instead, the first n minus one parts each take the floor and the final part carries the leftover, and the same total comes out.

Nothing here depends on where or when the distance was first run, on a world record, or on a training schedule. This page is about the length itself: one constant, six exact factors, three rounded descriptions and the arithmetic that connects them. A course that is measured short or long is a different question, and so is what a finish time would be over it. Both are left to other pages on this site, and this page says so rather than quietly answering them.

Every formula used on this page

The one constant

D = 42195 / 2 = 21097.5 m = 21.0975 km exactly. It is a fixed definition on this page, not an input and not a measurement of any one course. There is no distance field.

The six conversions

km = D / 1000; metres = D; mi = D / 1609.344; yd = D / 0.9144; ft = D / 0.3048; laps = D / 400. Every factor is exact, so the only rounding on the page is in the printed decimals.

Equal parts

For a part count n, per_m = D / n and share = 1 / n. The boundary after part k is k x per_m metres, and its share is k / n. The final boundary is n x per_m = D.

Whole-metre parts

whole = floor(D / n). The first n - 1 parts take whole metres each; the last takes D - (n - 1) x whole. The sum is D exactly.

13.1 miles

13.1 x 1609.344 = 21082.4064 m. Difference from the constant = 21097.5 - 21082.4064 = 15.0936 m short.

21.1 kilometres

21.1 x 1000 = 21100 m. Difference from the constant = 21097.5 - 21100 = -2.5 m, which is 2.5 m long.

13 miles 192.5 yards

13 x 1609.344 + 192.5 x 0.9144 = 20921.472 + 176.022 = 21097.494 m. Difference from the constant = 21097.5 - 21097.494 = 0.006 m, or 6 mm, short.

The exact imperial leftover

21097.5 - 13 x 1609.344 = 176.028 m. Divided by 0.9144 that is 192.506561679788 yd, which is what 192.5 rounds.

Whole-piece leftovers

21 x 1000 = 21000, so 21097.5 - 21000 = 97.5 m. 52 x 400 = 20800, so 21097.5 - 20800 = 297.5 m. Thirteen whole miles leave 176.028 m.

Display unit

per_display = per_m / factor(unit), where factor(km) = 1000, factor(mi) = 1609.344, factor(m) = 1, factor(yd) = 0.9144, factor(ft) = 0.3048 and factor(lap) = 400. The canonical value stays in metres; switching the unit reformats it and never reads the displayed number back.

Three worked examples

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

What this page deliberately does not do

• No prediction. This page does not read a recent race or tell you what you can run.
• No pace and no finish time. It does not turn a pace and a time into a distance, and it has no time field at all.
• No second race distance. One fixed constant, 21097.5 m, is the only standard on this page.
• No lap-by-lap split table, no per-kilometre checkpoint clock and no printable pace band.
• No course presets, no aid-station markers and no short-course or long-course time correction.
• No world records, no area records, no all-time lists and no history of the event.
• No age or sex grading, no sex selector, and no ability or finish-time label.
• No training plan, session prescription, weekly volume, coaching or medical advice.
• No elevation, hill, weather, heat, humidity or wind correction.
• No good-or-bad label, no cut-point and no grade for any length.
• No device sync, no GPS import, no account, no upload, no saved history and no server-side arithmetic.