Calculating slope from contour lines on a paper map

To find the slope between two points on a paper map, multiply the contour interval by the number of intervals between them to get the rise. Multiply the straight-line map distance by the scale denominator to get the horizontal run. Divide the rise by the run and multiply by 100 for a percentage grade. The method needs three readings and one division, and its accuracy depends on how well you read those three numbers.

Read the rise from the contour interval

The contour interval is the height difference between neighbouring contour lines. The USGS symbol sheet says the interval is printed in the margin of each USGS map, so check the legend or margin before you start. Count the contour steps between the two points. A point on the 1,200 m line and a point on the 1,260 m line are six intervals apart, so at a 10 m interval the rise is 60 m.

Contours never cross, and a point that lies between two lines has an elevation between their values. The rise from such a point is an estimate, and the error can be as large as one interval at that end. Measure from the nearest line where you can, and record in your notes that the figure is an estimate.

Measure the run on the map

The run is horizontal ground distance, so measure the straight line between the two points, not a curved route. Lay the edge of a strip of paper or a ruler between the points and read the length in millimetres. The conversion from map to ground uses the scale denominator, explained in the map scale guide.

run (m) = map distance (mm) × scale denominator ÷ 1000

Turn the rise and run into a grade and an angle

rise         = contour interval × intervals crossed
run          = map distance × scale denominator ÷ 1000
grade        = rise ÷ run × 100        (%)
angle        = arctan(rise ÷ run)      (°)
slope length = √(run² + rise²)         (m)

The grade and the angle describe the same slope in two units, and the grade and angle conversion sets out how to move between them. The slope length is the straight-line distance up the slope from one point to the other. It is always longer than the run.

rise         = 10 m × 6 = 60 m
run          = 12 mm × 25,000 = 300,000 mm = 300 m
grade        = 60 ÷ 300 × 100 = 20.0%
angle        = arctan(60 ÷ 300) = arctan 0.200 = 11.3°
slope length = √(300² + 60²) = √93,600 = 305.9 m

Check the result against the spacing of the lines

The spacing of the lines gives a cross-check. Over the 300 m run there are six intervals, so on an even slope the contours sit 300 ÷ 6 = 50 m apart on the ground. A 10 m rise over a 50 m spacing is 10 ÷ 50 × 100 = 20%, which matches the calculation. If the two disagree, look first for a misread interval or a mismeasured distance.

Measure a whole route in legs

A route rarely climbs at one grade. Split it into legs at each bend, saddle or change of slope, then work out the rise for each leg using the same method. Climbing legs and descending legs both count, but they go into different totals. Naismith’s rule and the trail time tools use the total ascent, which is the sum of the rises on the climbing legs only.

leg A: rise +60 m, run 300 m, grade +20%
leg B: rise −20 m, run 200 m, grade −10%
leg C: rise +40 m, run 250 m, grade +16%

total ascent  = 60 + 40 = 100 m
total descent = 20 m
net change    = 100 − 20 = +80 m (the finish sits 80 m above the start)

The total ascent is the figure to enter as the ascent in the trail time tool, because that tool reads climb and descent as separate totals. The net change tells you where the finish sits relative to the start, which matters for planning the return.

Keep the legs short where the slope changes sharply. A long leg that averages a gentle slope can hide a steep section at its start or end, and splitting it reveals the change. Short legs also make the measuring error larger in relative terms, so balance the two.

How a measuring error changes the grade

A 1 mm error on the map is 25 m on the ground at this scale. Against a 12 mm measurement, that is 1 part in 12, about 8.3% of the run, so the grade moves by a similar proportion. Measuring 11 mm gives a run of 275 m and a grade of 60 ÷ 275 × 100 = 21.8%. Measuring 13 mm gives a run of 325 m and a grade of 60 ÷ 325 × 100 = 18.5%. Both readings are within 1.8 percentage points of the 20.0% example.

A longer straight line between the two points reduces the relative effect of a fixed millimetre misreading. Where the terrain allows, pick two points that are further apart and still a known number of intervals apart.

Why the map shows an average

The grade is an average over the measured distance. A 20% average can hide a short step of 35% and a flat bench, and the map will not show either. Contours are drawn to show the general shape of the land, so a short steep step can fall between two lines and never appear.

Closely spaced contours represent steep slopes, and widely spaced contours or an absence of contours mean the ground is relatively level, according to the USGS symbol sheet. Read the spacing to see where the steep parts sit along a route, and use the calculation for the average over the whole stretch.

Walking speed also depends on the slope. The Tobler hiking function guide explains how grade sets walking speed, and the Naismith’s rule guide turns a distance and an ascent into a time for the whole route.

Where the method stops

The result depends on the survey behind the map and on the map’s age. Ground changes after a survey, with new tracks, cleared forest or slips, and a printed contour line will not show them. Readings also depend on how carefully the ruler is laid and how the interval was read.

The map slope tool runs the same steps for one leg at a time and also reports the slope length and a Tobler walking speed for that leg. Treat the speed as a model estimate for the leg you entered, not a measurement of how fast you will walk it.