How pitch is measured
Pitch is how many inches the roof rises for every 12 inches it runs horizontally. A 7/12 roof climbs 7″ over each foot. The angle, the percent grade and the multiplier are the same slope written three other ways.
Formula
- Angle
- arctan(rise ÷ 12)
- Grade
- rise ÷ 12 × 100
- Multiplier
- √(12² + rise²) ÷ 12
- Hip / valley
- √(rise² + 288) ÷ 12
Multiply a horizontal run by the multiplier to get the length along the slope. Multiply the common run by the hip / valley factor to get the length of a hip or valley where both roof planes have the same pitch. 288 is 12² + 12²: a hip crosses 12″ of run in each direction for every foot of common run.
The multiplier and the hip / valley factor are the first two lines of the rafter table on a framing square, divided by 12. At 7/12 the square reads 13.89 for a common rafter and 18.36 for a hip or valley, in inches per foot of common run.
Worked example
A 7/12 roof: arctan(7 ÷ 12) = 30.26°. The grade is 7 ÷ 12 = 58.3%. The multiplier is √(144 + 49) ÷ 12 = 1.1577, so a rafter that covers 10 ft of run is 11.58 ft long, or 11′-6 15/16″, before the overhang and plumb cuts. A hip over the same 10 ft of common run is 10 × 1.5298 = 15.30 ft.
Shingle slope limits
The IRC allows asphalt shingles on roofs of 2/12 or steeper. From 2/12 up to 4/12 the underlayment goes on in two layers; from 4/12 up, one layer is enough. Where the code map calls for wind design, the 2021 table asks for two layers at every slope.
The 2024 IRC keeps the same slope limits and adds a single layer of self-adhering membrane (ASTM D1970) as an alternative to two layers. Many towns adopt the IRC with amendments, so check with your building department, and follow the shingle maker's installation instructions.
References
- ICC, International Residential Code 2021: R905.2.2 slope limits for asphalt shingles; R905.1.1 and Table R905.1.1(2) underlayment application.
- ICC, International Residential Code 2024: R905.2.2 and Table R905.1.1(2).
- US Navy, NAVEDTRA 14044 Builder training manual, chapter 2, Roof Framing: rafter tables and worked rafter lengths.