# Tobler's hiking function

Tobler's hiking function – walking speed vs. slope angle chart.

Tobler's hiking function is an exponential function determining the hiking speed, taking into account the slope angle.[1][2][3] It was formulated by Waldo Tobler. This function was estimated from empirical data of Eduard Imhof.[4]

## Formula

Walking velocity:

${\displaystyle W=6e^{\displaystyle -3.5\left\vert {\frac {dh}{dx}}+0.05\right\vert }}$
${\displaystyle {\frac {dh}{dx}}=S=\tan \theta }$

where

W = walking velocity [km/h][2]
dh = elevation difference,
dx = distance,
S = slope,
θ = angle of slope (inclination).

The velocity on the flat terrain is 5 km / h, the maximum speed of 6 km / h is achieved roughly at -2.86°.[5]

On flat terrain this formula works out to 5 km/h. For off-path travel, this value should be multiplied by 3/5, for horseback by 5/4.[1]

## Pace

Pace is the reciprocal of speed.[6][7] For Tobler's hiking function it can be calculated from the following conversion:[7]

${\displaystyle p=0.6e^{\displaystyle 3.5\left\vert m+0.05\right\vert }}$

where

p = pace [s/m]
m = gradient uphill or downhill (dh/dx = S in Tobler's formula),

## Sample values

Pace in minutes per kilometer or mile vs. slope angle for Tobler's hiking function.
Slope
(deg)
(dh/dx)
Speed Pace
km / h mi / h min / km min / mi s / m
-60 -1.73 0.02 0.01 3603.9 5799.9 216.23
-50 -1.19 0.11 0.07 543.9 875.3 32.63
-40 -0.84 0.38 0.24 158.3 254.7 9.50
-30 -0.58 0.95 0.59 63.3 101.9 3.80
-25 -0.47 1.40 0.87 42.9 69.1 2.58
-20 -0.36 2.00 1.24 30.0 48.3 1.80
-15 -0.27 2.80 1.74 21.4 34.5 1.29
-10 -0.18 3.86 2.40 15.6 25.0 0.93
-5 -0.09 5.26 3.27 11.4 18.3 0.68
-2.8624 -0.05 6.00 3.73 10.0 16.1 0.60
0 0 5.04 3.13 11.9 19.2 0.71
1 0.02 4.74 2.94 12.7 20.4 0.76
5 0.09 3.71 2.30 16.2 26.0 0.97
10 0.18 2.72 1.69 22.1 35.5 1.32
15 0.27 1.97 1.23 30.4 49.0 1.83
20 0.36 1.41 0.88 42.6 68.5 2.56
25 0.47 0.98 0.61 60.9 98.1 3.66
30 0.58 0.67 0.41 89.9 144.6 5.39
40 0.84 0.27 0.17 224.6 361.5 13.48
50 1.19 0.08 0.05 771.8 1242.1 46.31