Gradients

Slope can be expressed in 3 ways:

  1. as an angle, eg. 10°, which is measured as the angle the ground makes with the horizontal;
  2. as a gradient, eg 1 in 10 or 1:10 or 1/10 or 0.10, which means that for every 10 units travelled horizontally, the ground rises (or falls) one unit vertically; and
  3. as a percentage, eg 10%, which is merely the gradient expressed as a fraction, and then converted to a percentage by multiplying by 100.

It is important to realise that the human eye (or rather the brain) invariably distorts slopes by exaggerating them. For example people will often judge a 1:3 slope to be a 1:1 slope. Similarly an slope of 15° will often be estimated to be close to 45°. This needs to be taken into account when you visualise slopes from a map!

Before examining how slopes can be calculated from maps, we will look briefly at converting gradients to slope angles, and vice versa.

Conversion of Gradients to % Slope

As mentioned above, the conversion of gradients to percentage slope, is very simple, either multiply (gradients to percentages) or divide (percentages to gradients) by 100. Thus a gradient of 1 in 10 is a 10% slope, and a 25% slope is a gradient of 25 / 100 or 1:4.

Example: Express a slope of 10° as a gradient.

To solve this problem you need elementary trigonometry: in particular, the rule that in a right-angled triangle, the tangent of an angle, is given by the length of the side of the triangle opposite the angle, divided by the length of the side adjacent to the angle.

tan A = o / a

In the following diagram you will see that the tangent of the slope angle S, is the vertical distance the ground rises for the horizontal distance travelled. Ie: that tan S = gradient

tan S = gradient

An angle of 12° = gradient of tan 12°.
Using a calculator this equals .2 (approx) or 1:5.
(This could also be expressed as a slope of 20%)

Click here to find answers (You need a calculator which can do tangents)

  1. Express a slope of 25° as a percentage.
  2. What is the angle of a 30% slope?

Calculating Slope at a Point on a Map

To calculate slope at a particular point on a map, first measure the distance between the two closest contours, convert this to a ground distance and then use the contour interval to calculate the gradient.

Example: On a 1:25,000 map, with a contour interval of 10m, the distance between contours is 2mm. What is the gradient?

2mm is equivalent to 2 x 25000mm ground distance = 50m
If the contour interval is 10m,
the gradient = 10:50
= 1:5

Click here to find answers

  1. In a particular area, the distance between contours on a 1:100 000 map, with a contour interval of 20m is 2mm. What is the gradient in this area?
  2. Roads and vehicle tracks should not exceed a slope of 1:3. On a 1:25000 map with a contour interval of 10m, what is the closest possible spacing of contours along a road.



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