Trigonometry on site: slopes, pitches, rise and run

Roofs, ramps, stairs, drains and setting out on site all come down to one shape: the right-angled triangle. Builders call its sides run (horizontal), rise (vertical) and the sloping side, for a roof, the rafter.

Pythagoras, sine, cosine and tangent

For a right-angled triangle with slope angle θ (measured from the horizontal):

1static double Rad(double deg) => deg * Math.PI / 180;2 3static double Rise(double run, double pitchDeg) => run * Math.Tan(Rad(pitchDeg));4static double Rafter(double run, double pitchDeg) => run / Math.Cos(Rad(pitchDeg));5 6double run = 4.0, pitch = 30;                 // half the span of an 8 m roof7Console.WriteLine($"Rise:   {Rise(run, pitch):F3} m");8Console.WriteLine($"Rafter: {Rafter(run, pitch):F3} m");

Rise:   2.309 m
Rafter: 4.619 m

Four ways to write a slope

The same slope is written differently by different trades, and converting between them is everyday work:

• Angle in degrees, from the horizontal: “a 30° roof”.
• Ratio rise : run, usually with the rise as 1: “a 1:12 ramp” rises 1 for every 12 along.
• Percentage, rise ÷ run × 100: “a 2% fall” on a flat roof or a drain drops 2 cm per metre.
• Pitch in inches of rise per 12 inches of run, common for roofs in the US: “a 6 in 12 roof”.

1static double PercentToDeg(double pct) => Math.Atan(pct / 100) * 180 / Math.PI;2static double RatioToPercent(double rise, double run) => rise / run * 100;3 4double ramp = RatioToPercent(1, 12);5Console.WriteLine($"1:12 ramp    = {ramp:F2}% = {PercentToDeg(ramp):F2} deg");6 7double roof = RatioToPercent(6, 12);8Console.WriteLine($"6 in 12 roof = {roof:F2}% = {PercentToDeg(roof):F2} deg");9 10Console.WriteLine($"100% slope   = {PercentToDeg(100):F2} deg");

1:12 ramp    = 8.33% = 4.76 deg
6 in 12 roof = 50.00% = 26.57 deg
100% slope   = 45.00 deg

Many accessibility codes, for example the US ADA standards, set 1:12 as the steepest allowed ramp, which is the 8.33% above. Check your local code: limits and landing rules vary by country.

Is the corner square? The cosine rule

Setting-out crews check a right angle with the 3-4-5 rule: measure 3 m along one side, 4 m along the other, and the diagonal must be exactly 5 m, because 3² + 4² = 5². If the diagonal is a little off, the cosine rule tells you the actual angle. It works for any triangle, not only right-angled ones:

c² = a² + b² − 2ab·cos C, so C = Acos((a² + b² − c²) / (2ab)), where C is the angle opposite side c.

1static double AngleDeg(double a, double b, double c)2    => Math.Acos((a * a + b * b - c * c) / (2 * a * b)) * 180 / Math.PI;3 4Console.WriteLine($"{AngleDeg(6, 8, 10):F2}");       // perfect 6-8-105Console.WriteLine($"{AngleDeg(6, 8, 10.012):F2}");   // diagonal 12 mm long

90.00
90.14

A 12 mm error on a 10 m diagonal means the corner is open by about 0.14°. Over a 30 m wall that is a 75 mm error at the far end: exactly the kind of number worth catching on the first day.

1double totalRise = 3000, maxRiser = 180, going = 250;2 3int risers = (int)Math.Ceiling(totalRise / maxRiser);4double riser = totalRise / risers;5double pitch = Math.Atan(riser / going) * 180 / Math.PI;6double run = (risers - 1) * going;7 8Console.WriteLine($"{risers} risers of {riser:F1} mm");9Console.WriteLine($"Going {going} mm, pitch {pitch:F1} deg");10Console.WriteLine($"Horizontal run {run} mm");

17 risers of 176.5 mm
Going 250 mm, pitch 35.2 deg
Horizontal run 4000 mm