Applications 11 min read
How to Calculate Roof Pitch: A Trigonometry Guide for Roofers
How to calculate roof pitch: tan(angle) = rise ÷ run, so a 6/12 roof is 26.57°. Pitch-to-angle conversions, rafter length, code minimums, a worked example.
Ramp Dimensions
Height from the lower surface to the upper surface
Fill this in to check the slope of an existing or space-limited ramp
Tip
Measure the rise with a level and tape from where the ramp will start, not from the bottom step: ground that falls away from the building adds rise.
Results
Run (horizontal)
-
Ramp surface length
-
Angle
-
Slope
-
Ratio (rise : run)
-
Straight footprint incl. landings
-
ADA slope check
-
Runs and landings (30 in max rise per run)
-
Ramp Profile
How ramp slope is calculated
A ramp is a right triangle lying on its side: the rise is the vertical leg, the run is the horizontal leg, and the walking surface is the hypotenuse. Slope is rise ÷ run, and the three ways of writing it (ratio, percent and degrees) are conversions of that one number.
run = rise × ratio (rise × 12 for a 1:12 ramp)
ramp length = √(rise² + run²)
angle = arctan(rise ÷ run)
slope % = rise ÷ run × 100
Worked example: a 24 in rise at 1:12
- Run = 24 × 12 = 288 in = 24 ft.
- Ramp surface = √(24² + 288²) = √83,520 = 289.0 in (24' 1").
- Angle = arctan(24 ÷ 288) = 4.76°; slope = 8.33%.
- The rise is under 30 in, so one run is enough. With 60 in landings top and bottom, the straight footprint is 288 + 120 = 408 in = 34 ft.
- The rise is over 6 in, so handrails are required on both sides.
Slope ratios compared
| Ratio | Angle | Percent | Run per 1 in of rise | Where it applies |
|---|---|---|---|---|
| 1:8 | 7.13° | 12.50% | 8 in | Existing buildings only, rise up to 3 in |
| 1:10 | 5.71° | 10.00% | 10 in | Existing buildings only, rise up to 6 in |
| 1:12 | 4.76° | 8.33% | 12 in | ADA maximum for new ramps |
| 1:16 | 3.58° | 6.25% | 16 in | Comfortable for manual wheelchairs |
| 1:20 | 2.86° | 5.00% | 20 in | Anything gentler is a walkway, not a ramp |
1:12 ramp sizes by rise
| Rise | Run | Ramp surface (ft) | Runs needed | Straight footprint with landings |
|---|---|---|---|---|
| 6 in | 6 ft | 6.02 | 1 | 16 ft |
| 12 in | 12 ft | 12.04 | 1 | 22 ft |
| 18 in | 18 ft | 18.06 | 1 | 28 ft |
| 24 in | 24 ft | 24.08 | 1 | 34 ft |
| 30 in | 30 ft | 30.10 | 1 | 40 ft |
| 36 in | 36 ft | 36.12 | 2 | 51 ft |
| 48 in | 48 ft | 48.17 | 2 | 63 ft |
| 60 in | 60 ft | 60.21 | 2 | 75 ft |
Footprint = run + a 60 in landing at the bottom, the top, and between runs. Switchback and L-shaped ramps need 60 × 60 in turning landings instead.
ADA ramp requirements at a glance
- Running slope: 1:12 (8.33%) maximum. Cross slope: 1:48 maximum.
- Rise per run: 30 in maximum before a level landing.
- Landings: top and bottom of each run, at least 60 in long and as wide as the ramp; 60 × 60 in where the ramp changes direction.
- Clear width: 36 in minimum between handrails.
- Handrails: both sides when the rise exceeds 6 in. Edge protection (a curb or extended surface) on open sides.
These figures come from section 405 of the 2010 ADA Standards for Accessible Design. The ADA applies to public and commercial facilities; for a private home your local residential code governs, and many jurisdictions adopt the same numbers. Confirm with your building department before you build.
The same triangle math drives the roof pitch calculator and the stair stringer calculator. To solve any other right triangle from two known values, use the right triangle calculator.
Frequently Asked Questions
How long does a wheelchair ramp need to be?
At the ADA maximum slope of 1:12 you need 12 inches of run for every inch of rise. A 24 in rise (about three steps) needs 288 in, or 24 ft, of sloped run. Add a 60 in landing at the top and bottom and the straight-line footprint is 34 ft.
What is a 1:12 slope in degrees and percent?
A 1:12 slope is arctan(1/12) = 4.76° and 1 ÷ 12 = 8.33%. Ratio, angle and percent all describe the same steepness: percent is rise ÷ run × 100, and the angle is the inverse tangent of rise ÷ run.
When does a ramp need a landing?
Under the 2010 ADA Standards (section 405) a single ramp run may rise no more than 30 in, which is 30 ft of run at 1:12. Taller ramps are split into runs with a level landing between them. Landings are also required at the top and bottom of every run and wherever the ramp changes direction, where they must be at least 60 × 60 in.
Is the ramp length the same as the run?
Not quite. The run is the horizontal distance; the ramp surface is the hypotenuse, √(rise² + run²). At 1:12 the difference is small (a 288 in run has a 289 in surface), but it grows quickly on steeper ramps such as loading or shed ramps, so order decking from the sloped length.
Can a ramp be steeper than 1:12?
Not for new ADA-covered construction. The ADA Standards allow 1:10 for a rise of up to 6 in and 1:8 for a rise of up to 3 in only in existing buildings where space is limited. Private homes are not covered by the ADA, but 1:12 is still the practical limit for an unassisted manual wheelchair user, and 1:16 to 1:20 is noticeably easier.
Do I need handrails?
ADA requires handrails on both sides of any ramp run with a rise greater than 6 in. They must be 34 to 38 in above the ramp surface and extend 12 in horizontally beyond the top and bottom of the run. Always confirm with your local building department, which may have stricter rules.
Related Guides
View all articlesDeepen your understanding of triangles and trigonometry with our guides and tutorials:
Applications 8 min read
Ladder Safety: The 4-to-1 Rule & Optimal 75° Angle Explained
The 4-to-1 ladder rule: 1 foot out for every 4 feet up sets a ladder at about 76°. The trig behind it, OSHA's version, placement examples, tipping risks.
Tutorial 11 min read
The Difference Between Square, Plumb, and Straight: A Guide for Builders and DIYers
Square, plumb and straight: what each term means, how to check it with a level, string line or the 3-4-5 method, worked numbers and typical tolerances.
Tutorial 9 min read
How to Square the Footings of a Building: A Complete Guide Using Trigonometry
How to square building footings with the 3-4-5 method and a diagonal check: the math, a worked 24 × 36 ft garage layout, tolerances, and layout tools.
Tutorial 8 min read
Trigonometry Word Problems: A Step-by-Step Solving Framework
Trigonometry word problems step by step: a five-step method (draw, record, identify, choose the ratio, solve and check) with worked examples and practice.
Applications 7 min read
Why Are Triangles the Strongest Shape? The Engineering Secret Hidden in Geometry
Triangles are the strongest shape in engineering: three fixed sides lock the angles. The geometry, a worked force example, and how trusses use it.