What roof pitch is, and why carpenters write it over 12
Roof pitch is the ratio of vertical rise to horizontal run, and in North American practice the run is fixed at 12 inches. A 6:12 roof climbs 6 inches for every foot you travel horizontally. Fixing the denominator at 12 is not arbitrary: a framing square has a 24-inch blade and a 16-inch tongue graduated in inches, so a carpenter can step off a rafter by holding 12 on the blade and the rise on the tongue, walking the square down the stock once per foot of run. The pitch is the layout instruction.
Everyone else in the trade wants a different form of the same number. A mitre saw wants degrees. A civil drawing, a gutter fall spec or a solar racking datasheet wants percent. A material estimator wants the slope factor, because roofing is sold by the sloped square while buildings are measured on the ground. A hip roof framer wants a second multiplier, because a hip rafter runs diagonally and climbs the same rise over a longer horizontal distance. All five numbers come from one right triangle, so converting between them is exact — there is no approximation and no table lookup you have to trust.
Note the vocabulary trap. Old texts use pitch to mean rise divided by the full span, so a 6:12 roof is a “quarter pitch”. Modern usage means rise over run. Everything on this page uses rise over a 12-inch run, which is what a pitch gauge, a plan callout and a shingle warranty all mean today.
The four conversions and where each one comes from
Start with the triangle. One leg is 12 (the run), the other is x (the rise), and the hypotenuse is the rafter line. Everything follows.
Angle. The tangent of the roof angle is opposite over adjacent, so θ = arctan(x/12). Going the other way, x = 12·tan(θ). This is the number you dial into a mitre saw for a plumb cut; the complementary angle, 90° − θ, is the seat cut.
Percent slope. Percent is just rise over run scaled by 100, so it equals 100x/12. A 12:12 roof is 100 percent slope, not 45 percent — percent slope and degrees only agree at zero. Confusing the two is the single most common unit error in roof drainage work.
Slope factor. The hypotenuse of the layout triangle is √(x² + 12²), and dividing by the 12-inch run gives the length of roof surface per unit of horizontal distance: SF = √(x² + 144)/12. Because that ratio is the same in every direction along the slope, it scales areas as well as lengths. Any roof plane's true area equals its shadow on the ground times its slope factor — which is why you can take off a whole roof from a tape measure at ground level using the roof area calculator.
Hip and valley factor. A hip rafter runs from a corner up to the end of the ridge, so in plan it travels diagonally: 12 inches across and 12 inches along for every foot of common run. Its horizontal travel per foot of common run is therefore √(12² + 12²) = 16.97 inches, and its true length per foot is √(x² + 288)/12. That is why a 6:12 hip works out to exactly 18 inches per foot of run — the tidiest number on the framing square — and it is the basis of the hip and valley rafter calculator.
Worked example: a 7:12 roof, by hand
You measure a rafter with a pitch gauge and read 7:12. The house footprint is 1,800 square feet including overhangs, and the common run is 14 feet.
- Angle. 7 ÷ 12 = 0.58333. arctan(0.58333) = 30.26°. Set the mitre saw to 30.3° for the plumb cuts and 59.7° for the seat cut, which is as fine as most saw scales read.
- Percent slope. 100 × 7 ÷ 12 = 58.33%.
- Slope factor. 7² = 49; 49 + 144 = 193; √193 = 13.8924; 13.8924 ÷ 12 = 1.1577.
- Sloped area. 1,800 × 1.1577 = 2,083.9 square feet, or 20.84 roofing squares before waste.
- Rafter line length. 14 ft of run × 1.1577 = 16.208 ft. The decimal 0.208 × 12 = 2.49 inches, so cut to 16 ft 2½ in from the ridge plumb cut to the heel of the birdsmouth.
- Hip factor. 49 + 288 = 337; √337 = 18.3576; ÷ 12 = 1.5298. A hip over the same 14-foot common run is 14 × 1.5298 = 21.42 ft, about 21 ft 5 in.
Every step here is reproducible with a scientific calculator and a tape. The only place people slip is step 4: they multiply the sloped area by the slope factor a second time, because they forgot the footprint was already a ground measurement.
How to read the number you get
Below 2:12, you do not have a shingle roof. The IRC sets 2:12 as the minimum slope for asphalt shingles, and between 2:12 and 4:12 it requires a doubled underlayment. Below 2:12 the assembly has to be a low-slope membrane — single-ply, modified bitumen or built-up. Manufacturer warranties track the same thresholds, so a 1.5:12 roof shingled anyway is both a code violation and an uninsured one.
4:12 to 9:12 is the ordinary residential band. Most tract housing sits at 4:12 to 6:12; steeper 8:12 to 12:12 roofs appear on Colonial, Tudor and Cape forms and on anything designed to shed heavy snow. Above about 7:12 crews normally stop walking the deck and set roof jacks, which shows up in labour pricing rather than material pricing.
The slope factor tells you what the pitch costs in material. Going from 4:12 to 12:12 raises the slope factor from 1.054 to 1.414, so the same footprint needs 34 percent more roofing. That number is exact and it is the honest way to compare two roof designs before you price them in the shingle bundle calculator.
Steeper roofs also carry less snow load but more wind uplift. ASCE 7 reduces the sloped-roof snow coefficient as pitch increases and warm roofs shed, which is why steep roofs are traditional in snow country; the roof snow load calculator applies that reduction. Do not use pitch alone as a structural conclusion.
Roof pitch conversion chart
| Pitch | Angle | Percent slope | Slope factor | Hip / valley factor | Rafter per ft of run |
|---|---|---|---|---|---|
| 1:12 | 4.76° | 8.3% | 1.0035 | 1.4167 | 12.04 in |
| 2:12 | 9.46° | 16.7% | 1.0138 | 1.4240 | 12.17 in |
| 3:12 | 14.04° | 25.0% | 1.0308 | 1.4361 | 12.37 in |
| 4:12 | 18.43° | 33.3% | 1.0541 | 1.4530 | 12.65 in |
| 5:12 | 22.62° | 41.7% | 1.0833 | 1.4743 | 13.00 in |
| 6:12 | 26.57° | 50.0% | 1.1180 | 1.5000 | 13.42 in |
| 7:12 | 30.26° | 58.3% | 1.1577 | 1.5298 | 13.89 in |
| 8:12 | 33.69° | 66.7% | 1.2019 | 1.5635 | 14.42 in |
| 9:12 | 36.87° | 75.0% | 1.2500 | 1.6008 | 15.00 in |
| 10:12 | 39.81° | 83.3% | 1.3017 | 1.6415 | 15.62 in |
| 11:12 | 42.51° | 91.7% | 1.3566 | 1.6853 | 16.28 in |
| 12:12 | 45.00° | 100.0% | 1.4142 | 1.7321 | 16.97 in |
Angles are rounded to two decimals; multipliers to four. Every value is θ = arctan(x/12), 100x/12, √(x²+144)/12 and √(x²+288)/12 evaluated at that pitch.
The code thresholds this page uses
The 2:12 minimum for asphalt shingles and the doubled underlayment requirement between 2:12 and 4:12 come from the International Residential Code, section R905.2.2, as adopted in the 2021 edition. Your jurisdiction may amend the IRC or adopt a different edition, and manufacturer instructions can be stricter than the code — where they are, the instructions govern. Check your adopted code and the shingle wrapper before you rely on a slope threshold.
Mistakes that put the pitch out
- Measuring the run along the roof. Run is horizontal. Lay the level flat, mark 12 inches out from the pivot, then measure straight down to the roof surface. On any pitched roof, measuring the 12 inches along the shingles instead of level gives a rise that is too small and a pitch that reads low — at a true 6:12 you would read 5.37:12.
- Confusing percent with degrees. A 100 percent slope is 45°, not 100° of anything. Drainage and solar documents almost always use percent; framing documents almost always use degrees or x:12.
- Measuring over shingle courses or a bowed rafter. Get the pitch off the underside of a rafter in the attic, or off a straight run of sheathing. Course lines are not a plane.
- Using the slope factor on a length that is already sloped. The factor converts plan to slope once, and only once. Apply it to a rafter length you already measured on the roof and that length is too long by the factor; every area you then build from it is too big by the same factor. On a 6:12 roof that is 11.8 percent of extra material you will pay for.
- Using the common slope factor on a hip or valley. The hip climbs the same rise over a longer run, so it needs √(x²+288)/12. On a 6:12 roof the difference is 1.500 against 1.118 — a third longer.
- Assuming both roof planes match. Additions, saltboxes and shed dormers routinely carry a second pitch.
Where this fits, and when to reach for another tool
Pitch conversion is the first step of nearly every roof calculation, not the whole of any of them. Once you have the slope factor, the roof area calculator turns a footprint and overhang into sloped area and squares. Once you have the angle and the run, the rafter length calculator gives line length, tail length and cut angles, and the birdsmouth calculator sizes the seat and heel cuts so the rafter bears properly on the plate.
What pitch does not tell you is whether the framing works. Rafter size, species, grade and spacing come from the span tables in IRC section R802 (or an engineered design), and they depend on snow load, dead load and whether the ceiling is attached. A steep roof with undersized rafters is still an undersized roof. For the members below it, use the floor joist span calculator and the relevant span tables rather than inferring anything from slope.
Outside North America, pitch is usually quoted directly in degrees or as a ratio over 1. The arithmetic is identical — substitute your run for the 12. Working from metric drawings, enter rise and run in millimetres or metres in measured mode and the calculator normalises them.
Terms used on this page
- Run
- The horizontal distance from the outside of the wall plate to a point directly under the ridge. For a symmetric gable it is half the span.
- Rise
- The vertical distance from the top of the wall plate to the top of the roof at the ridge, over that run.
- Slope factor
- The ratio of sloped length to horizontal length, √(x²+144)/12. It converts plan area to roof area and run to rafter length.
- Low-slope roof
- In IRC usage, a roof below 2:12, which must be covered with a membrane rather than shingles or tiles.