Angle, percent grade and roof pitch (x in 12) from rise and run or from any one of them, with the sloped length and rafter factor.
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This slope rises 0.5 for every 1 of run: 26.57° from horizontal, a 50 % grade, or a 6/12 roof pitch. Its sloped length is 13.4164 cm.
h is the rise, l the horizontal run, g the percent grade, p the rise per 12 of run and f the rafter factor.
Multiply a horizontal run by this to get the sloped length.
Every way of stating a slope comes from one ratio, m = rise ÷ run. The angle from horizontal is arctan m, the percent grade is 100m, the roof pitch is 12m (units of rise per 12 of run), and the rafter factor √(1 + m²) turns a horizontal run into the sloped length. Enter a rise and run, or any one of the other forms, and the rest follow.
The default rise of 6 over a run of 12 is a 6/12 roof: 26.57°, a 50% grade and a rafter factor of 1.1180, so 12 cm of run needs 13.4164 cm of rafter. Ramps and roads use the same numbers: the general maximum running slope in §405.2 of the US 2010 ADA Standards, 1:12, is an 8.33% grade, or 4.76°.
Percent grade is not an angle: a 100% grade is 45°, and a vertical face has no finite grade or pitch, so angles must lie between −90° and 90°. A negative rise describes a downhill slope.
A roof pitch of x/12 is a slope of m = x/12, and its angle is arctan m. A 6/12 roof has m = 0.5, so its angle is arctan 0.5 ≈ 26.57° and its grade is 50%. The same rise ÷ run ratio describes ramps, roads and the slope of a line.
All four are ways of writing one ratio, m = rise/run, with the run measured horizontally:
| Quantity | Formula from slope m | Meaning |
|---|---|---|
| Slope | m = rise/run | Vertical change per horizontal unit |
| Angle θ | θ = arctan(m) | Degrees from horizontal |
| Grade | 100m percent | Rise per 100 horizontal units |
| Roof pitch | 12m :12 | Rise per 12 horizontal units |
| Length factor | sqrt(1+m²) | Sloped length per horizontal unit |
To go back from an angle, use m = tan θ. A 30° roof has m = tan 30° ≈ 0.5774, a pitch of about 6.93/12 and a 57.7% grade.
Each line gives the angle, the percent grade and the rafter length factor:
Multiply the horizontal run by the length factor √(1 + m²). At 8/12 the factor is about 1.2019, so 10 ft of run needs a rafter about 12.02 ft long, before any overhang.
On a symmetrical gable roof, the run on each side is half the span: a building 24 ft wide has 12 ft of run per side.
For a line through (x₁, y₁) and (x₂, y₂), m = (y₂ − y₁)/(x₂ − x₁), the rise divided by the run. From (3, −2) to (8, 1), the rise is 3 and the run is 5, so m = 0.6: a 60% grade at about 30.963757°, or a 7.2/12 pitch.
OpenStax's linear-functions lesson derives this formula. For the equation of the line as well, use the coordinate geometry calculator.
Values are rounded to six decimal places.
| Example | Slope m | Angle ° | Grade % | Pitch :12 |
|---|---|---|---|---|
| default rise/run | 0.5 | 26.565051 | 50 | 6 |
| two-point differences | 0.6 | 30.963757 | 60 | 7.2 |
| downhill 3-4-5 | -0.75 | -36.869898 | -75 | -9 |
| flat | 0 | 0 | 0 | 0 |
| mixed feet/inches | 1 | 45 | 100 | 12 |
| one in twelve | 0.083333 | 4.763642 | 8.333333 | 1 |
The downhill row drops 3 over a run of 4. Its minus sign records the descent, while the length along the slope stays a positive 5. The mixed-units row is a rise of 1 ft over 12 in, equal lengths, so it is 45°.
One in twelve is the general maximum running slope for a ramp under §405.2 of the 2010 ADA Standards: an 8.33% grade, about 4.76°.
A flat line has slope 0 and a length factor of 1. A vertical face has zero run, so its slope, grade and pitch are undefined; angles must stay strictly between −90° and 90°.
Checked against: Independent calculation
Checked against: tan 45° = 1
Checked against: degrees(atan(0.08)); 12 × 0.08; 1/0.08
Checked against: degrees(atan(4/12)), sqrt(1 + 1/9)
Take the arctangent of the grade divided by 100: θ = arctan(grade/100). An 8% road grade is arctan 0.08 = 4.57°, and a 100% grade is 45°, not 90°. The reverse is grade = 100 × tan θ, so a 30° slope is a 57.7% grade.
26.57°. A pitch of x/12 means x units of rise for every 12 of run, so the angle is arctan(x/12): 4/12 is 18.43°, 6/12 is 26.57°, 8/12 is 33.69° and 12/12 is 45°. The rafter factor for 6/12 is √1.25 = 1.118, so 12 ft of horizontal run needs 13.42 ft of rafter before any overhang.
The general maximum running slope under the US 2010 ADA Standards (§405.2) is 1:12: one unit of rise for every 12 of run, an 8.33% grade or 4.76°. Each ramp run may rise at most 30 in (§405.6), so a 30 in rise needs at least 30 ft of horizontal run. Section 405.2 permits limited steeper slopes at existing sites where space is constrained: up to 1:10 for a rise of at most 6 in, or up to 1:8 for a rise of at most 3 in. Slope alone does not establish compliance; landings, width, cross slope and other requirements also apply. Other jurisdictions have their own building codes.
Divide the vertical rise by the horizontal run, both in the same unit: m = rise/run. A roof that rises 6 in over 12 in of run has m = 0.5. Measure the run horizontally, not along the surface: the sloped length here is 13.42 in, and dividing by it gives 0.447, the sine of the angle rather than the slope.
This calculator includes 8 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.