OpenSpecEngineering

Turns a rise and a run into the three common ways of stating a slope: percent grade, slope ratio and angle from horizontal. Surveyors, road and site designers, and builders use it to state the grade of a road, ditch, pipe, ramp, roof or embankment in whatever form a drawing or spec asks for.

Equations

G = 100·Δy/Δx     ratio = Δx/Δy : 1     θ = arctan(Δy/Δx)

Δy is the rise, the change in elevation, and Δx the run, the horizontal distance over which it happens [1]. G is the percent grade. The slope ratio is given as run to rise, so 3 : 1 means 3 units across for every 1 unit up, the usual way to state an embankment or ditch side slope. θ is the angle from horizontal in degrees. Rise and run can be in any length units, and each has its own unit picker.

Limits and common mistakes

• The run is the horizontal distance, not the distance measured along the slope. On a steep slope a taped or wheel-measured length is longer than the run, and the grade comes out too small. Correct it first: Δx = √(L² − Δy²).
• Percent grade is not the angle. A 100 % grade is 45°. For small slopes the angle in degrees is about 0.573 times the percent grade.
• Some fields, such as roofing and some structural drawings, give the ratio as rise to run (for example 1 : 12 or 4 in 12). The page gives run to rise, so 1 : 12 appears as 12 : 1.
• A negative rise is a downgrade. All three results then come out negative. A zero rise has no slope ratio, so the page rejects it.

Worked example

A new access road climbs 3.6 m between two stations 60 m apart in plan. The drawings need the grade in percent, and the grading crew wants the angle for its slope meter.

1. Enter Rise 3.6 m.
2. Enter Run 60 m and press Solve. The Solution shows: Percent Grade: 6 %
   Slope Ratio: 16.6667 : 1 (run : rise)
   Angle from Horizontal: 3.43363 deg

Check by hand: 3.6/60 = 0.06, or 6 %, and 60/3.6 = 16.67. The small-slope rule gives 0.573 × 6 = 3.44°, close to the exact arctan(0.06) = 3.434°.