Grade-Adjusted Pace Calculator

An uphill kilometre can take longer than a flat kilometre even when the runner is working just as hard. Raw pace records the difference in speed without separating the effect of the hill from the effort. Grade-adjusted pace (GAP) estimates a level-ground pace with the same modelled rate of energy use.

Because pace is time per unit distance, a smaller number means faster running. An uphill effort normally converts to a faster flat-equivalent pace. A gentle downhill effort converts to a slower one because the descent reduces the modelled energy needed per metre. Steeper descents do not keep becoming cheaper indefinitely, so downhill adjustments cannot be treated as the reverse of uphill adjustments.

Grade

The signed steepness of the slope, expressed as a percentage of vertical rise divided by horizontal distance. A +5% grade rises 5 metres over 100 horizontal metres. Negative grades describe descents. Percentage grade is not an angle in degrees.

Running distance

The distance along the sloping path, rather than the vertical height gained or the horizontal distance used to define grade. Pace describes the time needed to cover this running distance.

A sustained road climb, a steady hill repetition or an unchanging treadmill incline gives this comparison a useful setting. An undulating route does not have one representative slope. A loop can finish at its starting elevation while containing substantial climbs and descents, and those energy costs do not disappear when its net elevation change is zero.

Equal effort and equal total energy also describe different comparisons. A flat-equivalent time estimates how long the same distance would take at the modelled effort. An equal-energy flat distance estimates how far the energy spent on the original run would carry the runner on level ground. A longer equivalent distance does not add kilometres to the distance actually run.

These comparisons depend on a running-energy model, not a measurement of the runner's heart rate, fatigue or ability to sustain a predicted speed. They can help put a steady hill effort in context, but they do not establish a race result or a safe target pace. GAP is an informational exercise estimate, not medical advice or an individual training prescription.

How to Use This Tool:

Use the distance and moving pace or running time from one steady slope. Results update as you change the inputs.

1. Enter Run distance in kilometres or miles. Use the length of the running path, not elevation gain. Distance and pace units can be selected independently.
2. Choose Pace or Time under Enter as. For pace, enter a value such as 6:30 or 6.5 and choose min/km or min/mi. For time, enter the running duration over the whole distance, such as 32:30 or 0:32:30, and choose the Pace display basis. Exclude stops in either mode.
   Clock seconds must be below 60. An entry such as 6:60 is rejected rather than carried into the next minute; enter 7:00 for seven minutes.
3. Enter Actual grade as a positive uphill or negative downhill percentage. To compare the same effort on another slope, open Advanced and set Comparison grade. Leaving that grade at 0% gives the flat comparison.
4. Check Actual pace and Actual time in Pace comparison against the run you intended to enter. If the summary says Check your run inputs, correct the highlighted value. Grades must be from −45% to +45%, inclusive. An invalid comparison grade makes only the comparison unavailable; the flat-equivalent result remains available.

Interpreting Results:

Flat-equivalent pace and Flat-equivalent time describe the estimated level-ground pace and duration for the same run distance. Check the signed grade and pace unit before interpreting a large adjustment.

Cost ratio

The modelled energy cost per metre divided by its flat-ground cost. Above 1, the flat-equivalent pace is faster than actual pace. Below 1, it is slower. At 1, pace is unchanged. This is an energy-cost ratio, not a percentage improvement in speed.

Pace adjustment (flat − actual)

A negative adjustment means fewer seconds per kilometre or mile on flat ground. A positive adjustment means more seconds. Its percentage describes the change in time per unit distance, not the percentage change in energy cost.

Equal-energy flat distance

The modelled flat distance corresponding to the original run's energy expenditure. It is neither extra distance travelled nor a corrected measurement of the route.

Grade response compares 19 same-effort estimates from −45% to +45% in steps of 5 percentage points. It is not an elevation profile or a prescribed pacing plan. The chart follows your selected pace unit. The Run ledger uses seconds per kilometre for pace. Ledger distances use kilometres and total durations use seconds.

Technical Details:

Running energy cost measures the energy needed to move one kilogram of body mass through one metre of the running path. Multiplying that cost by speed gives the rate of energy use per kilogram. Holding this rate constant means a higher cost must be matched by a lower speed, or equivalently a longer time per unit distance.

Running energy cost and equal-effort pace

Let G be the signed Actual grade in percent. Its fractional slope i is the percentage divided by 100, so +5% becomes 0.05. The running-cost polynomial uses that fraction.

The result C is in joules per kilogram per metre, written J/kg/m. The flat reference is the fitted intercept of 3.6 J/kg/m, so the same fitted curve supplies both the sloped and flat costs.

For equal metabolic power, meaning the same modelled energy use per second, actual pace divides by the cost ratio to give flat-equivalent pace. A comparison slope then scales that flat pace by its own relative cost.

At an actual grade of 0%, the cost ratio is exactly 1 and pace is unchanged. A comparison grade equal to the actual grade returns actual pace; a comparison grade of 0% returns flat-equivalent pace. Body mass cancels in these ratios, so no body-weight input is needed. This does not provide a personal calorie total.

Multiplying pace in seconds per kilometre by distance in kilometres gives duration in seconds. Equal-energy flat distance instead multiplies the run distance by the cost ratio. The signed pace adjustment and its percentage compare flat pace with actual pace.

Here Δp is the pace adjustment in s/km, and A is the displayed adjustment percentage. For fixed actual pace and grade, changing distance changes the durations and equal-energy distance, but not GAP. In time-entry mode, changing distance also changes the actual pace derived from the running time.

Pace conversion and numerical precision

One mile is exactly 1.609344 kilometres. Multiply a distance entered in miles by that factor before using D. Let M be pace in decimal minutes per selected distance unit, and T be total running time in decimal minutes. The three input paths all produce actual pace in seconds per kilometre.

Clock entries are converted to minutes before these equations are used. Seconds need a two-digit integer part and must be below 60; in an hours:minutes:seconds entry, the middle minutes must also be two digits from 00 to 59. Fractional seconds are accepted, such as 6:30.5. Equivalent clock and decimal entries produce the same calculation.

Calculations retain full numerical precision. Clock-formatted paces and durations round to the nearest second, after any conversion to the selected pace unit. Other numeric readings show up to three decimal places. Total times use unrounded paces, so multiplying a displayed pace by distance can differ from the displayed total time.

Accepted values

All endpoints below are inclusive. A time entry is accepted only when its duration divided by distance gives a pace within the same pace limits.

These distance and pace bounds are input limits, not a range of scientifically validated running performances. Estimated paces are not restricted to that input interval, so a mathematically valid result can still be unattainable.

Evidence and assumptions

The Minetti study published in 2002 examined 10 runners in treadmill trials spanning −45% to +45%. Its Figure 1 gives the running-cost regression used here. This laboratory evidence describes running energy cost; converting that cost into equal-effort pace adds the steady-power assumption below.

The pace conversion assumes steady running and equal metabolic power rather than measuring effort directly. One fixed cost curve is applied without adjustments for technical terrain, wind, fatigue, individual running economy or descending skill. Walking and sprinting are outside its intended scope. The calculation handles one steady grade, not a sequence of route segments, and does not reproduce Strava's proprietary GAP. The study supports the energy-cost relationship; it does not validate every pace or finish time that can be generated from it.

Worked Examples:

References:

• Energy cost of walking and running at extreme uphill and downhill slopes, Minetti et al., Journal of Applied Physiology, September 2002. Figure 1 contains the running-cost regression.