How the common rafter calculation works
A common rafter, the run and the rise make a right triangle. The run is the level side, the rise is the vertical side and the rafter is the sloped side. Pitch is written as x in 12: the roof rises x inches for every 12 inches of run. Everything follows from that:
- Rafter length per 12 in of run = √(x² + 12²). At 6 in 12 that is √180 = 13.416 in.
- Slope factor = √(x² + 12²) ÷ 12. Multiply any level distance by it to get the distance along the slope.
- Rise = run × x ÷ 12.
- Roof angle = arctan(x ÷ 12) in degrees. 12 in 12 is 45 degrees.
- Ridge deduction = half the ridge board thickness × slope factor. The rafters on each side stop short of the ridge centerline by half the board, measured level, so the cut length is shorter by that distance along the slope.
- Cut length = (run + overhang) × slope factor − ridge deduction.
Run is half the building span, from the outside face of the wall to the ridge centerline. Overhang is measured level from the same wall line, not along the rafter tail. Mixing a level overhang with a sloped one is the most common way to cut a tail wrong.
Cut angles: plumb and seat
The plumb cut is the vertical cut at the ridge and at the tail. The seat cut is the level cut where the rafter sits on the wall plate. If you set a saw from a square cut across the board, the plumb cut is the roof angle off square and the seat cut is 90 degrees minus the roof angle. At 6 in 12 that is 26.57 degrees for the plumb cut and 63.43 degrees for the seat cut. On a framing square, you read the same cuts by lining up x and 12 along the rafter edge.
Roof pitch chart
These values come from the same formulas the calculator uses. The length column is the rafter length in inches for every foot of run, before any overhang or ridge deduction.
| Pitch | Angle | Rafter per ft of run | Plumb cut off square | Seat cut off square |
|---|---|---|---|---|
| 3 in 12 | 14.04° | 12.369 in | 14.04° | 75.96° |
| 4 in 12 | 18.43° | 12.649 in | 18.43° | 71.57° |
| 5 in 12 | 22.62° | 13.000 in | 22.62° | 67.38° |
| 6 in 12 | 26.57° | 13.416 in | 26.57° | 63.43° |
| 7 in 12 | 30.26° | 13.892 in | 30.26° | 59.74° |
| 8 in 12 | 33.69° | 14.422 in | 33.69° | 56.31° |
| 9 in 12 | 36.87° | 15.000 in | 36.87° | 53.13° |
| 10 in 12 | 39.81° | 15.620 in | 39.81° | 50.19° |
| 12 in 12 | 45.00° | 16.971 in | 45.00° | 45.00° |
Worked example 1: a 24 ft wide house at 6 in 12
Span 24 ft, so the run is 12 ft (144 in). Pitch 6 in 12, overhang 12 in level, 1.5 in ridge board. These are invented study numbers.
- Slope factor = √180 ÷ 12 = 1.1180.
- Rise = 144 × 6 ÷ 12 = 72 in (6 ft). Angle = arctan(0.5) = 26.57 degrees.
- Run along the slope = 144 × 1.1180 = 161.00 in. Overhang along the slope = 12 × 1.1180 = 13.42 in.
- Ridge deduction = 0.75 in × 1.1180 = 0.84 in.
- Cut length = 161.00 + 13.42 − 0.84 = 173.57 in, which is 14 ft 5 9/16 in to the nearest 1/16 in.
Worked example 2: 21 ft wide at 8 in 12 with a longer overhang
Span 21 ft, so the run is 10 ft 6 in (126 in). Pitch 8 in 12, overhang 16 in, 1.5 in ridge board.
- Slope factor = √208 ÷ 12 = 1.2019. Angle = arctan(8 ÷ 12) = 33.69 degrees.
- Rise = 126 × 8 ÷ 12 = 84 in (7 ft).
- Level distance used = 126 + 16 − 0.75 = 141.25 in. The 0.75 is half the ridge board, measured level.
- Cut length = 141.25 × 1.2019 = 169.76 in, which is 14 ft 1 3/4 in.
Taking the 0.75 off before multiplying and taking 0.90 off after give the same answer. Forgetting the slope factor on the deduction would leave the rafter about 0.15 in long. These are original arithmetic examples, not exam questions.
What this means on an inspection
You do not cut rafters, but plans give a pitch as x in 12 and spans as level dimensions, and a tape laid along a rafter reads the sloped length. The sloped length is always longer than the run. Rafter spans in the IRC tables are measured along the horizontal projection, not the sloped length, as the span table guide walks through. Check in your own book, at R802.4.1, how your edition measures the span, and convert with the slope factor before you compare.
Pitch also changes the thrust at the walls. For the same load, a flatter roof pushes outward harder than a steep one. That is why where the tie sits matters, and the rafter ties vs collar ties post shows what to look for. Roof and ceiling construction carries 14% of the B1 outline, and the outline guide shows where roof framing sits among the other areas.
Rafter length questions
How do I find the pitch of a roof I can reach?
Hold a level against the underside of a rafter, mark 12 inches along the level, and measure straight down from that mark to the rafter. That drop is x in 12. The calculator's measured rise option takes any rise and level run, such as 8 in over 24 in, which is 4 in 12.
Is run the same as span?
No. Span is the full width between the outside faces of the walls. Run is half of that for a symmetrical roof, to the ridge centerline. Typing the full span as the run doubles the answer.
Why take half the ridge board off the rafter?
The rafters on the two sides meet the ridge board on its faces, not at its centerline. Each rafter stops half the board's thickness short of the centerline. That half thickness is level, so you scale it by the slope factor before taking it off the sloped length.
Common rafter framing is part of the roof area of the B1. The calculator covers the geometry, and the guides cover where the code sets what is allowed.