Slope and Grade Converter (%, Degrees, 1:12, Roof Pitch)
Slope and Grade Converter (%, Degrees, 1:12, Roof Pitch)
Convert a slope between per cent grade, degrees, a 1 in n ratio, millimetres per metre, inches per foot and roof pitch in twelfths, in every direction — with the published limits that make the number mean something: the ADA’s ramp rules, the IRC’s minimum roof slopes and two countries’ drainage falls.
Slope, grade, pitch and fall
A 1:12 ramp — the ADA maximum running slope — entered as a per cent grade
Four linear notations, and one arctangent
- rise/run
- the slope itself, a dimensionless number. Everything else on the page is this, rescaled or arctangented
- grade %
- rise over run as a percentage. 100% is 45°, and there is no upper bound
- theta
- the angle. The one quantity here that is NOT proportional to the slope, which is why you cannot interpolate in it
- pitch x : 12
- the roofer’s notation: inches of rise per foot of run, so x = 12 × rise/run
- length factor
- how much longer the sloping surface is than its plan. The number to order materials against
Worked example
A 1:12 ramp — the ADA maximum running slope — entered as a per cent grade
1 in 12 means one unit of rise for twelve of run, so the slope is 1/12 = 0.083333 and the grade is 8.3333%
The angle is arctan(1/12) = 4.7636° — note that it is not 8.33° and not 1/12 of 90°
The same slope is 83.33 mm per metre, 1.000 inches per foot and a roof pitch of 1.00 in 12
Over the 10 m run in the box that is a rise of 0.833 m, along a sloping length of 10.0347 m — 0.347% more surface than the plan
And the ADA's other two rules bite: § 405.6 allows a 30 inch rise per run, so this ramp may be 9.144 m long before it needs a 60 inch landing, and § 405.3 limits the cross slope to 1:48
Five notations for the same slope, side by side
| Per cent | Degrees | Ratio 1 : n | mm per metre | in per foot | Roof pitch x : 12 | Where you meet it |
|---|---|---|---|---|---|---|
| 1.000 | 0.5729 | 1 : 100.000 | 10.0 | 0.1200 | 0.120 | A long, gentle road climb; the IPC’s 1/8 in per foot drain is just above it |
| 2.000 | 1.1458 | 1 : 50.000 | 20.0 | 0.2400 | 0.240 | The IRC’s minimum for most flat-roof coverings, which the code calls “2 percent” |
| 2.083 | 1.1935 | 1 : 48.000 | 20.8 | 0.2500 | 0.250 | The ADA maximum cross slope, and the IPC minimum for a small drain |
| 2.500 | 1.4321 | 1 : 40.000 | 25.0 | 0.3000 | 0.300 | Approved Document H’s 1:40 foul drain |
| 5.000 | 2.8624 | 1 : 20.000 | 50.0 | 0.6000 | 0.600 | The ADA’s threshold: gentler than this and it is a walk, not a ramp |
| 8.333 | 4.7636 | 1 : 12.000 | 83.3 | 1.0000 | 1.000 | The ADA maximum ramp running slope |
| 10.000 | 5.7106 | 1 : 10.000 | 100.0 | 1.2000 | 1.200 | A steep street; the limit for most wheelchair-accessible vehicle ramps |
| 16.667 | 9.4623 | 1 : 6.000 | 166.7 | 2.0000 | 2.000 | A 2:12 roof — the IRC’s minimum for asphalt shingles, with double underlayment |
| 20.833 | 11.7683 | 1 : 4.800 | 208.3 | 2.5000 | 2.500 | A 2½:12 roof — the IRC’s minimum for clay and concrete tile |
| 25.000 | 14.0362 | 1 : 4.000 | 250.0 | 3.0000 | 3.000 | A 3:12 roof, which the IRC calls “25 percent” and which is exactly right |
| 33.333 | 18.4349 | 1 : 3.000 | 333.3 | 4.0000 | 4.000 | A 4:12 roof, the IRC’s minimum for slate, which the code calls “33 percent” |
| 50.000 | 26.5651 | 1 : 2.000 | 500.0 | 6.0000 | 6.000 | A 6:12 roof — the classic domestic pitch |
| 100.000 | 45.0000 | 1 : 1.000 | 1,000.0 | 12.0000 | 12.000 | A 45° slope. Not vertical. This is the row people get wrong |
Accessibility ramps — 2010 ADA Standards for Accessible Design, United States
| Section | Requirement | As a ratio | Per cent | Degrees | mm per metre | Note |
|---|---|---|---|---|---|---|
| 405.2 | Ramp running slope, maximum | 1 : 12 | 8.333 | 4.7636 | 83.3 | Steeper than 1:20 is what makes a walking surface a ramp (106.5). |
| 405.3 | Ramp cross slope, maximum | 1 : 48 | 2.083 | 1.1935 | 20.8 | Measured across the direction of travel. |
| 403.3 | Walking surface running slope, maximum, before it is a ramp | 1 : 20 | 5.000 | 2.8624 | 50.0 | At 1:20 or gentler it is a walk and needs no handrails or landings. |
| 405.2 Ex. 1 | Existing site, rise 3 in maximum | 1 : 8 | 12.500 | 7.1250 | 125.0 | An alteration-only exception, not a design target. |
| 405.2 Ex. 2 | Existing site, rise 6 in maximum | 1 : 10 | 10.000 | 5.7106 | 100.0 | Also alterations only. |
Minimum roof slopes by covering — International Residential Code, United States
| Section | Covering | Minimum pitch | Per cent | Degrees | Ratio 1 : n | Note |
|---|---|---|---|---|---|---|
| R905.2.2 | Asphalt shingles | 2.00 : 12 | 16.67 | 9.462 | 1 : 6.00 | 2:12 to 4:12 requires double underlayment. |
| R905.3.2 | Clay and concrete tile | 2.50 : 12 | 20.83 | 11.768 | 1 : 4.80 | |
| R905.4.2 | Metal roof shingles | 3.00 : 12 | 25.00 | 14.036 | 1 : 4.00 | The code calls this 25 percent. |
| R905.5.2 | Mineral-surfaced roll roofing | 1.00 : 12 | 8.33 | 4.764 | 1 : 12.00 | The code calls this 8 percent. |
| R905.6.2 | Slate and slate-type shingles | 4.00 : 12 | 33.33 | 18.435 | 1 : 3.00 | The code calls this 33 percent. |
| R905.7.2 | Wood shingles | 3.00 : 12 | 25.00 | 14.036 | 1 : 4.00 | |
| R905.8.2 | Wood shakes | 3.00 : 12 | 25.00 | 14.036 | 1 : 4.00 | |
| R905.9.1 | Built-up roof | 0.25 : 12 | 2.08 | 1.193 | 1 : 48.00 | Coal-tar built-up may go to 1/8:12. |
| R905.10.2 | Metal panels, lapped seams, no sealant | 3.00 : 12 | 25.00 | 14.036 | 1 : 4.00 | |
| R905.10.2 | Metal panels, lapped seams with sealant | 0.50 : 12 | 4.17 | 2.386 | 1 : 24.00 | |
| R905.10.2 | Standing-seam metal panels | 0.25 : 12 | 2.08 | 1.193 | 1 : 48.00 | |
| R905.11.1 | Modified bitumen | 0.25 : 12 | 2.08 | 1.193 | 1 : 48.00 | |
| R905.12.1 | Thermoset single-ply | 0.25 : 12 | 2.08 | 1.193 | 1 : 48.00 | |
| R905.13.1 | Thermoplastic single-ply | 0.25 : 12 | 2.08 | 1.193 | 1 : 48.00 | |
| R905.14.1 | Sprayed polyurethane foam | 0.25 : 12 | 2.08 | 1.193 | 1 : 48.00 |
Drainage falls, with the jurisdiction named
| Jurisdiction | Document | Application | As published | Per cent | Ratio 1 : n | mm per metre |
|---|---|---|---|---|---|---|
| United States | IPC Table 704.1 | Horizontal drain 2½ in or smaller | 1/4 in per ft | 2.0833 | 1 : 48.0 | 20.83 |
| United States | IPC Table 704.1 | Horizontal drain 3 in to 6 in | 1/8 in per ft | 1.0417 | 1 : 96.0 | 10.42 |
| United States | IPC Table 704.1 | Horizontal drain 8 in or larger | 1/16 in per ft | 0.5208 | 1 : 192.0 | 5.21 |
| England and Wales | Approved Document H, Table 1 | Foul drain 75 mm, flow under 1 L/s | 1:40 | 2.5000 | 1 : 40.0 | 25.00 |
| England and Wales | Approved Document H, Table 1 | Foul drain 100 mm, at least one WC | 1:80 | 1.2500 | 1 : 80.0 | 12.50 |
| England and Wales | Approved Document H, Table 1 | Foul drain 150 mm, at least five WCs | 1:150 | 0.6667 | 1 : 150.0 | 6.67 |
Five notations, and the one that is not linear
Five trades, five notations, one quantity — and one of the five does not behave like the others. A percentage grade, a ratio of one in n, millimetres per metre, inches per foot and a roof pitch of x in twelve are all the same number rescaled: double the slope and every one of them doubles. The angle in degrees does not. It is the arctangent, and that single fact is behind the commonest mistake in the subject.
A 100% grade is 45°. Not 90°, not vertical, not “as steep as it gets”. Per cent grade is rise over run times a hundred, which is the tangent of the angle times a hundred, and at 45° the rise equals the run. A 200% grade is a real slope — 63.4° — and a vertical face is an infinite percentage. Halfway up, the error is at its worst: 50% is 26.57° where a linear reading of the scale would say 45. The chart on this page draws the true curve against the straight line people assume, and the gap between them is the size of the mistake.
The numbers that make a slope actionable are published, and they belong to particular jurisdictions. That is the reason this page has four tables rather than one conversion. In the United States, § 405.2 of the 2010 ADA Standards for Accessible Design caps a ramp’s running slope at 1:12, which is 8.33% and 4.764°. Two parts of that rule get left out of every conversion table: § 405.6 limits a single ramp run to a 30 inch rise, so at 1:12 you get 9.144 m of ramp and then a landing at least 60 inches long, and § 405.3 caps the cross slope separately at 1:48. A 1.5 m rise done properly is three runs and two landings, not one very long ramp.
Roofs are a covering question rather than an aesthetic one. The International Residential Code sets a minimum slope for each covering, because the real question is whether water will sit long enough to find a way under a lap. Asphalt shingles: 2:12 at the absolute minimum, and below 4:12 the code wants two layers of underlayment. Clay and concrete tile: 2½:12. Slate: 4:12. Wood shingles and shakes: 3:12. Standing-seam metal, which behaves like a membrane with raised joints rather than a set of overlapping pieces: a quarter in twelve. The code’s own percentages are rounded, incidentally — it calls a quarter in twelve “2 percent” when it is 2.08, and 4:12 “33 percent” when it is 33.33 — and 3:12, which it calls 25 per cent, is the one that happens to be exact.
Drainage has two codes on this page and they use incompatible units. The IPC’s Table 704.1 speaks in inches of fall per foot of run: a quarter for pipe up to 2½ inches, an eighth for 3 to 6 inches, a sixteenth for 8 inches and above. Approved Document H, for England and Wales, gives bare ratios: 1:40 for 75 mm, 1:80 for 100 mm with at least one WC on it, 1:150 for 150 mm with at least five. Both are converging on the same physics, and both get gentler as the pipe grows, because a big pipe carries a shallower stream and too much fall lets the water outrun the solids it is supposed to be carrying.
And the number nobody asks for until they are ordering materials. A sloping surface is longer than its plan. The factor is the square root of one plus the slope squared: a 4:12 roof is 5.4% more area than the drawing shows, and a 12:12 roof is 41.4% more. This page computes it for whatever run you enter, along with the rise over that run and the run needed for a given rise.
For angles as coordinates rather than as gradients, the DMS to decimal degrees converter handles the three formats and what each decimal place is worth on the ground.
Frequently asked questions
Is a 100% grade a vertical slope?
No. A 100% grade is 45°, and this is the single commonest error in the subject. Per cent grade is the rise divided by the run, times a hundred — which is the tangent of the angle, times a hundred. At 45° the rise equals the run, so the tangent is one and the grade is 100%. There is no upper limit to a percentage grade: a 200% grade is 63.4° and a vertical face is an infinite percentage. The conversion is not linear, and the chart on this page shows exactly how far the linear assumption drifts.
How do I convert per cent grade to degrees?
Take the arctangent of the grade divided by a hundred. So 8.33% is arctan(0.0833) = 4.764°, and 25% is 14.036°. Going the other way, multiply the tangent of the angle by a hundred. The only thing to be careful of is that every other notation on this page — ratios, millimetres per metre, inches per foot, roof pitch — is linear in the slope, and the angle is not, so you cannot interpolate between degrees.
What is a 4:12 roof pitch in degrees?
18.43°, which is 33.33% and a ratio of 1:3. Roof pitch is written as the rise over a run of twelve because twelve inches is a foot, so a 4:12 roof rises four inches per foot. The IRC makes 4:12 the minimum for slate and slate-type shingles, and calls it “33 percent”, which is the code rounding its own arithmetic.
What is the maximum slope for a wheelchair ramp?
In the United States, 1:12 — 8.33%, 4.76° — under § 405.2 of the 2010 ADA Standards for Accessible Design. But the running slope is only half the requirement: § 405.6 caps a single ramp run at a 30 inch rise, so at 1:12 you get 9.14 m of ramp and then you need a landing at least 60 inches long. The cross slope is limited separately to 1:48. And anything gentler than 1:20 is not a ramp at all under the standard, which is why 1:20 turns up as a design target. Other countries set their own figures; do not assume 1:12 travels.
What is the minimum fall for a drain?
It depends on the pipe and on which country’s code you are under, which is why the table names the jurisdiction. The International Plumbing Code Table 704.1 asks for a quarter inch per foot — 1:48, 2.08% — for horizontal drains 2½ inches and smaller, an eighth of an inch per foot for 3 to 6 inch pipe and a sixteenth for 8 inch and larger. England and Wales work from Approved Document H Table 1, which gives 1:40 for a 75 mm foul drain, 1:80 for a 100 mm drain serving at least one WC and 1:150 for a 150 mm drain serving at least five. Gentler is not always better and neither is steeper: too much fall and the water outruns the solids.
What is the difference between grade, slope, pitch and fall?
Mostly the trade that is speaking. Grade is a percentage and comes from roads and railways. Slope is a ratio or a percentage and comes from civil engineering and surveying. Pitch is x:12 and comes from roofing, where the twelve is a foot. Fall is a ratio, or inches per foot, and comes from plumbing and paving. They are the same quantity and this page converts between all of them, but the words are not interchangeable in a specification — and one genuine distinction survives: in some roofing literature “pitch” means the rise over the whole span rather than over the half-span, which halves the number, so read carefully before you cut a rafter.
How much extra roofing material does a slope need?
The square root of one plus the slope squared, minus one. A 4:12 roof has a slope of 0.333, so the sloping surface is 1.0541 times the plan area — 5.4% more. A 12:12 roof needs 41.4% more. The page computes it for whatever run length you give it, which is the number to order materials against; the plan area is the number a drawing gives you.
Related calculators
References
- United States Department of Justice, 2010 ADA Standards for Accessible Design, Chapter 4: § 405.2 ramp runs shall have a running slope not steeper than 1:12; § 405.3 cross slope not steeper than 1:48; § 405.6 the rise for any ramp run shall be 30 in (760 mm) maximum; § 405.7.3 landing clear length 60 in (1 525 mm) minimum; § 405.7.4 a 60 by 60 in landing where the ramp changes direction; § 106.5 a ramp is a walking surface with a running slope steeper than 1:20. Verified against the identically worded 2012 Texas Accessibility Standards text as published by Corada.
- International Code Council, International Residential Code, Chapter 9, Roof Assemblies. Minimum slopes as cited in the table: R905.2.2 asphalt shingles 2:12, with double underlayment from 2:12 to 4:12; R905.3.2 clay and concrete tile 2½:12; R905.4.2 metal roof shingles 3:12; R905.5.2 mineral-surfaced roll roofing 1:12; R905.6.2 slate 4:12; R905.7.2 wood shingles and R905.8.2 wood shakes 3:12; R905.9.1 built-up roofs ¼:12, coal tar ⅛:12; R905.10.2 metal panels 3:12 lapped without sealant, ½:12 with sealant, ¼:12 standing seam; R905.11.1 to R905.15.1 modified bitumen, thermoset and thermoplastic single-ply, sprayed polyurethane foam and liquid-applied coatings ¼:12. A model code adopted with amendments state by state.
- International Code Council, International Plumbing Code, Table 704.1, Slope of Horizontal Drainage Pipe: ¼ in per ft for pipe 2½ in and smaller, ⅛ in per ft for 3 to 6 in, 1/16 in per ft for 8 in and larger. The code text is behind a subscription and could not be fetched for this batch; the figures are as quoted by secondary sources citing Table 704.1 and should be checked against the adopted edition before they are relied on. The Uniform Plumbing Code takes a different approach, defaulting to ¼ in per ft for all sizes with a reduction to ⅛ available for 4 in and larger by permission.
- HM Government, Approved Document H: Drainage and waste disposal (England and Wales), Table 1, as reported by Local Authority Building Control: a 75 mm foul drain at 1:40 for flows under 1 L/s, a 100 mm drain at 1:80 where it serves at least one WC, and a 150 mm drain at 1:150 where it serves at least five. Cited through LABC because the Approved Document itself was not fetched for this batch.
- The conversions themselves need no citation, but they were checked rather than assumed: every notation on this page is round-tripped through every other at ten slopes from 0.5% to 200%, and each cited code limit is reproduced in all five notations, in _c02b_proof.py.
- National Institute of Standards and Technology. NIST Handbook 44, Appendix C, General Tables of Units of Measurement. The US liquid gallon is 231 cubic inches exactly and the international inch is 25.4 mm exactly; every volume on this page descends from those two definitions.