Slope Calculation from Contour Lines in a TopographicMap

Slope is the measure of steepness or the degree of inclination of a feature relative to the horizontal

plane. Gradient, grade, incline and pitchare used interchangably with slope. Slope is typically

expressed as a percentage, an angle, or a ratio. The average slope of a terrain feature can conveniently

be calculated from contour lines on a topo map. To find the slope of a feature, the horizontal distance

(run) as well as the vertical distance (rise) between two points on a line parallel to the feature need to be

determined. The slope is obtained by dividing the rise over run. Multiply this ratio by 100 to express

slope as a percentage. The slope angle expressed in degrees is found by taking the arctangent of the

ratio between rise and run.

Here we want to find the

average slope of the face of this mountain (the section from point A to point B). The vertical distance or

rise is the elevation difference between point A and point B. Checking the topo map below Point A is at

2500m. Contour interval is 20m (five contour lines per 100m elevation difference). Therefore elevation of

point B is 2780m. Rise = 2780 - 2500 = 280m.

The run or the horizontal distance between two points is found by using the map's scale bar. Using a ruler

we can measure the scale bar of Google Maps at bottom left corner. 17mm or 1.7cm on the map is equal

to 100m in the real world. Again using a ruler, the next step is measuring the horizontal distance between

point A and point B on the map: 42mm or 4.2cm. Calculating the real world distance: Run = 4.2cm *

(100m / 1.7cm) = 247m. (Note that the numbers corresponding to measurements on the image may be

different on your computer monitor due to resolution difference or when the image is printed. The end

result however should be the same).

 

If no ruler is available for exact measurements, you can always use any straight-edge object or the side of

your finger. Mark the length of the scale bar on the straight edge, and compare the scale bar length to the

distance along the line between two points on the map. See how many scale bar lenghts or its portion will

equal the distance between the points. An approximate direct distance measurement will be obtained

conveniently. For example here the distance between points A and B is about two and a half times the

distance on scale bar that corresponds to 100m. Therefore the desired map distance is approximately 2.5

x 100m = 250m.

 As noted in the map scale and distance measurement section, the distance - bearing and the line drawing

tools of the Geokov Map Maker  can be used to directly measure distances on digital maps.

Gradient (decimal) = Rise / Run =

280m / 247m = 1.1336

Here for every 1 unit (e.g. meter, foot, etc.) of horizontal travel, there is 1.1336 units of altitude gain.

 Alternatively for every 0.882 unit horizontal travel, there is one unit of vertical gain. Therefore as a ratio,

the gradient would be expressed as (1 in 0.882).

Gradient (percentage) = 1.1336 * 100 = 113.4%

Slope angle is the angle α in the diagram. By definition of tangent in trigonometry: tan α = Rise / Run

Therefore having the values for rise and run, value of α in degrees is determined by taking the arctangent

 

(tan-1) of the ratio: α = arctan(280/247) = 48.6° 

The distance of travel (distance along the slope or hypotenuse of triangle) is obtained from the

Pythagorean theorem equation: (hypotenuse distance)2 = 2472 + 2802. Distance along the slope is equal

to 373m.

So keep in mind that the distances found on map using the map's scale are the horizontal distances

(straight distance as the crow flies). If you plan to do a long traverse in mountainous terrain, your actual

travel distance will be much longer than the one calculated with the scale bar. To get a more real distance

of travel, the distances along slopes will need to be taken into account as above.  Inclinometers

(clinometers) are used for direct slope measurements in field work such as forestry, terrain mapping,

avalanche safety, etc. Many modern compasses include inclinometers.

Note: top down (2D) satellite map with overlaid contour lines using Geokov Map Maker  is used here for

clarity. Exactly the same procedures as above are used with a regular topographic map to find the

desired parameters.

The above value is the meseasure of the average slope for the distance between two points. In reality

parts of the slope are steeper and parts gentler than the average slope. The terrain most often is not a

naturally smooth surface and there are usually sections with varying slopes. Also features such as cliffs,

convex rolls, knolls, dips, benches, etc. that are smaller in size than the contour interval might not show

up on the topo map. For example you may be travelling on a 35° slope and come across a 10m cliff band,

while the topo map shows a constant 30° to 40° slope. Terrain assessment and getting a feel for the

slope from contour lines (without measuring) comes with experience. Sometimes measuring sections of

the whole slope is required, such as in avalanche terrain mapping where slope angles for the start zone,

track and run-out zones are measured seperately and a profile for the avalanche path is drawn.

 

Some slope features are important in field

studies like geomorphology, avalanches and backcountry travel decision making. Examples

include convexand concave slopes. Convex slopes roll from less steep to steeper terrain. Depending on

the contour interval and the size of the feature, convexities on terrain may be detected by wider contour

spacing on top and closer contour lines on the bottom of the

roll. Concave slopes go from steeper to

gentler terrain with movement downslope. There are closer contour spacing at the top and wider spacing

at the bottom of the concave slope.