Slope [mil.wndsn.com]

Measuring Slope

Level: Basic

Instrument: Wndsn NATO-MIL Telemeter

The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. Grade is a special case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of tilt. Often slope is calculated as a ratio of "rise" to "run," or as a fraction (rise over run) in which "run" is the horizontal distance and "rise" is the vertical distance.

Figure 1. Elements in slope calculation.

To measure slope, we have various options on our quadrant. They all start with aligning one of the edges of the device with the slope to be measured. To read the result, we can choose between 4 scales:

The degree scale on the Quadrant
The percent scale on the Quadrant

Measuring in degrees

Degrees measure the angle of inclination to the horizontal. This is the angle θ opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run. The tangent of θ is equal to the rise divided by the run. Therefore, the inverse-tangent of the rise divided by the run will give the angle:

tan(θ) = rise / run    
    θ  = arctan(rise / run)

Measuring as percentage

Percentage, is the result of the formula:

100 × rise / run

which could also be expressed as:

100 × tan(angle of inclination)

Once we have a given slope value in percent, we can convert to degrees and vice versa, by using the two adjacent scales.

Examples

Figure 2. Measuring slope in degrees: 19° on the transversal scale, measured parallel to the short edge of the Telemeter. Note the quadrant scale showing 90 - 19°.

Figure 3. Measuring slope in percent, parallel to the long edge of the Telemeter: ~3° or 5%.

Let rise = 1 m, run = 20 m.

Degrees:

tan (θ) = (1/20)    
     θ  = arctan(1/20)    
     θ  = 2.86°

Percentage:

Slope = 1/20
      = 0.05 × 100
      = 5%

See also:

NATO-MIL Telemeter