Basics of Slope Calculations (SD - HD - Elevation)

 

Slope Measures - Units

Slope is a measure of steepness.  Units can be in degrees, percent or as a ratio. 

Degrees: Most of us are familiar with slopes measured in degrees.  There are 360 degrees in a full circle.  From the perspective of traversing we are essentially interested in measures between 0° and 90°.  A measure of 0° indicates flat ground and a slope of 90˚ is essentially a vertical line (i.e. perpendicular to the horizon, or "straight up").  A slope of 45˚ is exactly half way between the previous two measures.  With a 45˚ slope, a movement of 10 meters horizontal means you also moved 10 meters vertical (see diagram below).

 

 

Text Box: 45°

 

Text Box: 10 m
Text Box: 10 m

 

 

  

We can see that for every step forward along the 45˚ slope, an equal increment is made both horizontal and vertical.

Slopes can also be expressed as a ratio or percent.  The calculation is the familiar "rise / run"

As a ratio:

In the case above we have a rise and a run of 10 m each.

    Slope = rise / run  = 10m / 10m = 1.0

 

If we had a rise of 3 m and a run of 10 m, then

    Slope = rise / run  = 3m / 10m = 0.3

 

You should note that rise/run is the same as opposite/adjacent ... which is the same as tangent.  Tangent is a ratio, it expresses vertical rise as a ratio (or "%") of horizontal distance.  (This will be important later when we want to calculate change in elevation from our traverse notes).  You can determine the 'tangent ratio' is two ways"

- if lengths are known, then use rise / run (= opposite / adjacent    = elevation change / horizontal distance)

- if angle is known, then simply use the tan button on your calculator (it simply converts degrees to a ratio)

 

As a percent:

To convert a ratio to a percent we simply multiply by 100.  Expressed as a percent the first measure would be 100% (1.0 * 100) and the second measure would be 30% (0.3 * 100).

Thus the angle in the diagram above can be expressed as 45˚ or 1.0 (ratio) or 100%.

 

Slope Measures - Instrument

To measure slope inclination in the field we use a clinometer (Suunto is the most common make).  This device provides measures in both degrees and percent.  The numbers on the left side are in degrees and the numbers on the right are in percent.  (You can always double check this by 'looking up' in clinometer until the reading is 45 on one side and 100 on the other (the 100 indicating the % side).

 

Converting Slope Measures Between Percent and Degrees

Sometimes we need to be able to convert slope percent to degrees ...

Arctan (or inverse tangent) is the opposite of tangent.  Remember tangent converts an angle to a ratio - well ... arctan converts a ratio to an angle. Forget all the mathematical proofs … to convert from slope ratio (or percent) to slope degrees you will use Arctan.  Please do not be afraid (“step away from that Course Drop Form”) – it is simply a button on your calculator that will magically convert slope percent to degrees.  Just so you know, 32 degrees is equivalent to 63% slope (try looking into your clinometer to check this).  To practice on your calculator ... punch “0.63” into your calculator, then punch the button(s) for “arctan” – you will get ~32 (degrees).  So, the equation to convert slope percent to degrees is:

Slope degrees = arctan (slope percent)but remember you need to enter slope percent as a ratio, thus 63% is entered as 0.63

 

Convert SD to HD

When we traverse in the 'real world' we take measures of slope distance (SD) and steepness (slope %).  However, in order to map features in their proper place we need to convert SD to HD (horizontal distance). 

Measures of slope in degrees are useful in converting slope distance to horizontal distance.  Remember cosine is “adjacent over hypotenuse”.  If you consider “adjacent” = horizontal distance and “hypotenuse” = slope distance, then cosine = HD / SD.  Think of cosine as the ratio of horizontal distance to slope distance.  If this ratio (i.e. cosine) = 0.85, then for every 1 m of slope distance you are actually traveling only 0.85 m horizontal.  A slope of 32° has a cosine of 0.85.  (Try 'plugging' 32 into your calculator and then press the COS button - you should get 0.848).

Converting SD to HD: Consider that you have traveled 25 m slope distance with a slope angle of 32º.  What is the horizontal distance? 

 HD    = SD * cosine (slope degrees)

= 25 m * cosine(32)

= 25 m * 0.85

= 21.2 m horizontal

 Now remember that we record slopes in percent, not degrees.  So we will need to first convert slope percent to slope degrees.  As per the previous section, we Arctan to accomplish this.

Example: our raw field measures are: SD = 25 m and slope = 63% ... calculate HD.  Stepwise we

  1. convert slope % to degrees - to do this we take the arctan of the ratio (i.e. for 63% we take the arctan of 0.63)

  2. take the cosine of the degrees to get a ratio (of HD to SD) ... this number will 'reduce' (convert) SD to HD

  3. multiply this ratio by SD to get HD

In equation form ...

HD   = SD * cosine * [arctan * (slope ratio)]

Our raw data is SD =25 m, and slope = 63% ... (last time our slope was in degrees, this time it is in the conventional percent)

HD    = 25m * cos [arctan (0.63)]

        = 25m * cos [ 32º ]

        = 25m * 0.85

        = 21.2 m

 

Convert HD to SD

Sometimes we need to solve for HD.  This is often the case when we want to establish plots at fixed intervals (e.g. 100 m grid).  These intervals are typically in HD.  To solve for slope distance, the above equation is simply rearranged:

SD      = HD / cosine [arctan (slope ratio)]

using the same data, assume you need to go 21.2 m horizontal distance to get to plot centre and the slope angle was 63% …

SD      = 21.2 m / cosine [arctan (0.63)]

          = 21.2 m / cos [ 32º ]

          = 25 m

  

Determine change in elevation

Measures of slope in degrees are useful for converting slope distance to horizontal distance, but percent is easier to use to calculate change in elevation.  Remember that tan is slope is expressed as a ratio ... "rise / run".  In other words, rise (elevation change) expressed as a ratio (%) of horizontal distance.  The equation is simply ...

Elev. Change    = HD * slope ratio

In our example the raw data was SD =25 m, and slope = 63%.  We calculated HD to be 21.2 m.  Thus to calculate elevation change ...

                            = HD * slope ratio

                            = 21.2 m * 0.63

                            = 13.2 m

 Thus, for a SD of 25 m @ 63% slope you traveled 21.2 m horizontal and had an elevation change of 13.2 m.

 

SUMMARY

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Slopes can be in degrees, %, or ratio (TAN)

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We record slopes in % (e.g. 63%)

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We use slope ratios in our calculations (e.g. 0.63)
 

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We convert slopes to degrees in order to convert between SD and HD

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ARCTAN converts slope ratio to degrees

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slope degrees = arctan * (slope ratio)
 

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We use cosine to convert SD to HD

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HD   = SD * cosine * [arctan * (slope ratio)]

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HD   = SD * cosine * slope degrees
 

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We rearrange this formula to convert HD to SD

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SD      = HD / cosine [arctan (slope ratio)]
 

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We use slope ratio directly to determine elevation change

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Elev. Change    = HD * slope ratio