How route gradient is calculated

The basic calculation

Percentage gradient compares vertical change with horizontal distance.

For a simple straight slope, divide the rise by the horizontal run and multiply by 100. A route that gains 10 metres over 100 horizontal metres has a 10 per cent gradient. A descent of the same shape can be written as minus 10 per cent when direction matters.

Both values must use the same unit. Ten metres divided by 0.1 kilometres works only after one has been converted. The calculation is dimensionless, which is why the final result can be expressed as a percentage.

Some route systems use distance measured along the line rather than a strictly horizontal run. Ordnance Survey distinguishes planimetric length from length along a three-dimensional surface in its path-network specification. The difference is small on gentle slopes and grows with steepness. Product documentation should state which distance supports its gradient.

Three notations

Ten per cent, one in ten and ten degrees are not the same slope.

A ratio of 1 in 10 describes one unit of vertical change for ten units of horizontal run. That is 10 per cent. The corresponding angle is found with trigonometry and is about 5.7 degrees, not 10 degrees. The Health and Safety Executive uses the same distinction when warning that a 10-degree road is considerably steeper than a 10 per cent road.

Percentage and ratio can therefore be converted directly when both use horizontal run. Converting to degrees requires the arctangent of rise divided by run. On a map or profile, avoid estimating that angle from the line on screen because the horizontal and vertical axes are usually drawn at different scales.

The averaging window

A gradient is always a gradient over some distance.

A kilometre-long climb may average 5 per cent while containing a short ramp that is much steeper. Calculate only from its endpoints and the ramp disappears into the average. Calculate between every pair of noisy GPS points and tiny height errors can create implausible maximums.

Route software addresses this by choosing a segment length, smoothing height, resampling distance or combining those steps. A short window reveals local changes but is more sensitive to noise and horizontal misalignment. A long window is steadier but generalises brief features. There is no useful maximum-gradient comparison unless the products use comparable windows and elevation sources.

Switchbacks add another complication. The path may gain height gradually over a long travelled distance while the terrain rises sharply in a shorter straight-line direction. Use the route's own distance, not the fall line of the hillside, when judging effort along the path.

A number with a location

Find the segment before deciding what the percentage means for the journey.

• LengthSeparate a sustained climb from a very short change between samples.
• DirectionThe same physical slope is an ascent one way and a descent the other.
• SurfaceA paved rise and a loose technical trail can demand different pace and skill at the same gradient.
• ApproachPlacement after earlier climbing can matter more than the route maximum alone suggests.

Use the beacen Guide for current route controls.

Read each percentage beside its profile, map and activity information. Gradient describes change over a selected interval; it is neither a safety grade nor a complete difficulty score.