Pace, speed and time are one relationship read three ways

Pace and speed feel like different quantities because they are quoted in different shapes — one as minutes per kilometer, the other as kilometers per hour. They are the same relationship written upside down from each other, and seeing that once removes most of the confusion around them.

One relationship, three readings

There is a single fact underneath: a distance, a time, and how they relate. Fix any two and the third follows.

• Pace is time divided by distance: min / km.
• Speed is distance divided by time: km / h.
• Time is pace multiplied by distance.

Pace and speed are reciprocals with a unit change bolted on, which gives the most useful identity in the whole subject: pace in min/km = 60 / speed in km/h. Twelve kilometers an hour is 60 / 12 = 5 minutes per kilometer. Ten is 60 / 10 = 6. Fifteen is 60 / 15 = 4.

The free Running Pace tool takes a distance in kilometers and a time in minutes and prints all three readings. Its worked default is 10 kilometers in 55 minutes: pace is 55 / 10 = 5.5 minutes per kilometer, shown as 5:30 because 0.5 × 60 = 30 seconds, and speed is 10 / (55 / 60) = 10.909091 kilometers per hour. Check the reciprocal: 60 / 10.909091 = 5.5.

Kilometers to miles, and the number to memorize

The tool converts the distance first, using 1 km = 0.621371 mi, so 10 kilometers is 6.21371 miles and the mile pace is 55 / 6.21371 = 8.851395 minutes, printed as 8:51 because 0.851395 × 60 = 51.08 seconds.

The same answer comes from working on the pace directly. A mile is longer than a kilometer, so it takes longer to cover, so the mile pace is the bigger number: multiply by 1.609344, which is exact — a mile is defined as exactly 1609.344 meters. Here 5.5 × 1.609344 = 8.851392.

Those two routes differ in the sixth decimal place, and the difference is instructive rather than a bug. The tool's 0.621371 is a rounded reciprocal: 1 / 1.609344 = 0.621371 only to six places, and continues 0.6213711922. Round a factor and you inherit a rounding. At 8.851392 against 8.851395 it is three millionths of a minute, both print as 8:51, and nobody will ever run the difference.

That factor is worth carrying around as add roughly sixty percent, and the arithmetic is easy enough to do while moving:

• 4:00 per km is 4 × 1.609344 = 6.437376 minutes, which is 6:26.
• 5:00 per km is 8.04672 minutes, which is 8:03.
• 6:00 per km is 9.656064 minutes, which is 9:39.

The direction is the thing people get backwards. If a figure gets smaller when you convert kilometers to miles, the conversion went the wrong way.

Equal steps in speed are not equal steps in pace

Because pace is one divided by speed, a fixed increase in speed does not buy a fixed improvement in pace. Going from 10 to 11 kilometers an hour moves pace from 6 minutes per kilometer to 60 / 11 = 5.454545, which is 5:27 — a saving of 0.545455 minutes, about 32.7 seconds per kilometer. Going from 15 to 16 moves pace from 4 to 3.75, which is 3:45, a saving of 15 seconds.

Same one kilometer an hour, less than half the effect on the clock. That is why pace charts look compressed at the fast end, and why speed and pace give different impressions of the same change.

Small pace changes compound over distance

The last piece is the one that makes pace worth thinking about at all: pace multiplies by distance, so a difference too small to feel per kilometer becomes a visible difference in total time.

Five seconds per kilometer is nothing to sense directly. Over a marathon distance of 42.195 kilometers it is 5 × 42.195 = 210.975 seconds, which is 3 minutes and 31 seconds. Ten seconds per kilometer over the same distance is 421.95 seconds, just over 7 minutes. Over a half marathon of 21.0975 kilometers, five seconds per kilometer is 105.4875 seconds, about 1 minute 45 seconds.

Run it the other way and the same multiplication answers the planning question. A target of four hours over the marathon distance is 240 / 42.195 = 5.687878 minutes per kilometer, or 5:41, which in miles is 5.687878 × 1.609344 = 9.153752 minutes, or 9:09. Whether that number is reachable is not an arithmetic question and this page does not pretend otherwise — but what it costs per kilometer is, and it is one division.

The whole thing in one line

Distance, time, pace and speed are four names for two independent numbers. Write down the two you know, divide in whichever direction gives the unit you want, and remember that 60 divided by one of them gives the other. Everything else, including every figure above, follows from that.