Short answer
Tobler’s hiking function gives walking speed on a uniform slope: W = 6 × e^(−3.5 × |S + 0.05|) km/h, where S is the rise divided by the horizontal run. Flat ground gives 5.04 km/h. The peak, 6 km/h, sits at S = −0.05, a gentle 2.9° descent. A 10% climb gives 3.55 km/h, and a 10% descent gives 5.04 km/h, the same as flat.
Tobler’s hiking function gives walking speed on a uniform slope from one input, the slope S. The speed is W = 6 × e^(−3.5 × |S + 0.05|) km/h, where S is the rise divided by the horizontal run. On flat ground S is zero and W is 5.04 km/h. The speed peaks at 6 km/h when S is −0.05, a gentle descent of 2.9°.
The formula and its terms
The function is an exponential curve with an absolute value in the exponent. The term |S + 0.05| sets where the peak falls and how quickly speed drops away on either side of it. S is the change in height divided by the distance walked, which Wikipedia writes as dh/dx = S = tan θ, where θ is the angle of the slope. A 10% climb has S = +0.10 and θ = 5.7°. A 10% descent has S = −0.10 and θ = −5.7°.
W = 6 × e^(−3.5 × |S + 0.05|) W in km/h; S = rise ÷ run, negative downhill flat (S = 0): 6 × e^(−3.5 × 0.05) = 6 × e^(−0.175) = 6 × 0.8395 = 5.04 km/h
According to Wikipedia, the function was estimated from empirical data published by the Swiss cartography professor Eduard Imhof. It is therefore a fitted curve over slopes, not a law of human movement, and the speeds below are the curve’s values.
Speed at each slope
The table covers slopes from 30% downhill to 40% uphill. The last column is 60 ÷ speed, which turns the speed into minutes for each kilometre.
| S (rise ÷ run) | Angle | Speed (km/h) | Minutes per km |
|---|---|---|---|
| −0.30 | −16.7° | 2.50 | 24.0 |
| −0.20 | −11.3° | 3.55 | 16.9 |
| −0.15 | −8.5° | 4.23 | 14.2 |
| −0.10 | −5.7° | 5.04 | 11.9 |
| −0.05 | −2.9° | 6.00 | 10.0 |
| 0 (flat) | 0.0° | 5.04 | 11.9 |
| +0.05 | 2.9° | 4.23 | 14.2 |
| +0.10 | 5.7° | 3.55 | 16.9 |
| +0.15 | 8.5° | 2.98 | 20.1 |
| +0.20 | 11.3° | 2.50 | 24.0 |
| +0.30 | 16.7° | 1.76 | 34.0 |
| +0.40 | 21.8° | 1.24 | 48.3 |
The table is a convenience. The function is continuous, so a slope between two rows is read from the formula itself. A slope of +0.12, which is a 12% climb, has |0.12 + 0.05| = 0.17, so W = 6 × e^(−3.5 × 0.17) = 6 × e^(−0.595) = 6 × 0.5516 = 3.31 km/h. That falls between the 3.55 km/h at +0.10 and the 2.98 km/h at +0.15, as it should.
Speeds on slopes of the same size but opposite sign are not the same. A climb of 15% and a descent of 15% differ by 2.98 km/h against 4.23 km/h, which is why a route’s total ascent and total descent cannot be swapped for each other without changing the time.
Why the peak sits on a descent
The function is not symmetric about zero. It is symmetric about S = −0.05, so speed falls at the same rate on either side of that point. A 10% descent (S = −0.10) gives 5.04 km/h, the same as flat ground, because |−0.10 + 0.05| = |0 + 0.05| = 0.05. A 10% climb (S = +0.10) gives 3.55 km/h, so it takes 16.9 min per kilometre against 11.9 min on the flat, 1.42 times as long.
The reason is the shape of the curve, not a claim about walkers. The model treats a very gentle descent as the fastest slope, and everything steeper, up or down, as slower.
Off-path travel
The function describes travel on a path. Wikipedia states that for off-path travel the value should be multiplied by 3/5. On flat ground that gives 5.04 × 3/5 = 3.02 km/h, so one flat kilometre off path takes 60 ÷ 3.02 = 19.9 min, against 11.9 min on a path. The factor is a single multiplier, so it does not separate smooth ground from rough ground on the same slope.
Comparing with Naismith’s rule on a 10% climb
Take 1 km of horizontal distance with 100 m of climb, a 10% grade. Naismith’s metric rule gives 1 km ÷ 5 km/h = 12 min for the distance and 100 m ÷ 100 × 10 min = 10 min for the climb, 22 min in all. That is an effective speed of 1 km ÷ (22 ÷ 60) h = 2.73 km/h. Tobler’s function gives 1 km ÷ 3.55 km/h = 16.9 min on the same slope. The two results differ by about 5 minutes.
The difference reflects what each model describes. Tobler gives an instantaneous speed on one uniform slope and nothing else. Naismith gives a time for a whole route, with a fixed climb allowance built in. Neither includes rests. For a route with a mix of slopes, convert each section with one model and keep to that choice. The Naismith’s rule guide covers the route-level arithmetic, and the Trail time tool gives the Naismith figure for this exact leg.
Measuring the slope on a map
The Map slope tool takes the measurements from a map and computes the grade, the angle, the slope length and the Tobler speed for the leg. The example uses a 1:25,000 map and a leg going uphill.
Worked example: a 10% climb on a 1:25,000 map
Measured: 20 mm between two points on the map, and 5 contour intervals of 10 m crossed going up.
ground distance = 20 mm × 25,000 ÷ 1,000 = 500 m rise = 10 m × 5 = 50 m grade = 50 ÷ 500 × 100 = 10%, so S = +0.10 slope length = √(500² + 50²) = 502.5 m speed = 6 × e^(−3.5 × |0.10 + 0.05|) = 6 × e^(−0.525) = 3.55 km/h leg time = 0.5 km ÷ 3.55 km/h = 0.141 h = 8.45 min
The leg time is for the walking speed on that slope only. It has no rests, no allowance for footing and no change for a pack, so the tool should be read as one input to a plan.
What the function leaves out
The function models speed on a uniform slope. It has no term for the surface, load, fitness, fatigue or rests. Real slopes are rarely uniform, so a route should be split into legs before the function is applied. A short steep pitch and a long even grade with the same average slope can feel very different on foot, which is one more reason to work leg by leg.
Load matters as well. A heavier pack lowers the speed a walker can hold, and the function does not model that. The Pandolf equation guide covers the energy side of carrying a load on a grade. For how the slope percentage relates to the angle, see grade percent vs degrees.
The planning step, from a leg speed to a time to turn back, is covered in turnaround time. Slope readings from a contour map are covered in slope from contour lines.
Common questions
What is walking speed on a 10% uphill slope?
Tobler’s function gives 3.55 km/h on a 10% climb: W = 6 × e^(−3.5 × 0.15) = 6 × 0.5916 = 3.55 km/h. Off path, the 3/5 factor gives about 2.13 km/h. These are model values for a uniform slope, not a timed walk.
Is walking downhill faster than walking on the flat?
Only on a gentle slope. The peak is 6 km/h at S = −0.05, a 2.9° descent. Flat ground gives 5.04 km/h, and steeper descents are slower again: at S = −0.30 (16.7°) the model gives 2.50 km/h.
Why does this differ from Naismith’s rule?
Tobler gives speed on one uniform slope, with no rests. Naismith gives the time for a whole route, using a fixed climb allowance. Compare the two by converting the same leg with each model, as the 10% example on this page does.
Sources
Published 2 November 2026. Reviewed 2 November 2026. Sources support the figures quoted; they do not endorse Trailwright. Report a correction through the owner’s contact page.