Formula
A common rafter is a right triangle. The run is half the span, minus half the ridge. The rise comes from the pitch. The line length is the hypotenuse, along the top of the rafter.
run = span ÷ 2 − (ridge thickness ÷ 2) ÷ 12
rise = run × (pitch ÷ 12)
line = √(run² + rise²)
tail = overhang ÷ cos(arctan(pitch ÷ 12))
The pitch angle is arctan(pitch / 12). The seat cut is the complement. This is the triangle in every carpentry text (Wagner, Modern Carpentry) and the one the span tables assume. It is not a hip or a valley. Pitch in degrees, percent, or a measured rise and run is the roof pitch calculator.
Worked examples
24 foot span, 6/12, no overhang, no ridge. Run = 12 ft. Rise = 6 ft. Line = √(144 + 36) = √180 = 13.4164 ft, or 13 ft 5 in.
Same roof, 1½ inch ridge and 1 foot of overhang. Run = 12 − 0.0625 = 11.9375 ft. The tail is 1 / cos(26.565°) = 1.118 ft. The cut length is the line plus that tail.
Reference
24 foot span, no overhang, no ridge board. Lengths are the line length, to the nearest eighth of an inch.
| Pitch | Angle | Line length | Decimal feet |
|---|---|---|---|
| 3/12 | 14.04° | 12 ft 4 3/8 in | 12.369 ft |
| 4/12 | 18.43° | 12 ft 7 3/4 in | 12.649 ft |
| 5/12 | 22.62° | 13 ft 0 in | 13 ft |
| 6/12 | 26.57° | 13 ft 5 in | 13.416 ft |
| 7/12 | 30.26° | 13 ft 10 3/4 in | 13.892 ft |
| 8/12 | 33.69° | 14 ft 5 1/8 in | 14.422 ft |
| 9/12 | 36.87° | 15 ft 0 in | 15 ft |
| 10/12 | 39.81° | 15 ft 7 1/2 in | 15.62 ft |
| 12/12 | 45° | 16 ft 11 5/8 in | 16.971 ft |
Questions
How long is a common rafter?
Take half the span, subtract half the ridge thickness, and treat that as the run. Rise is the run times the pitch divided by 12. The line length is the hypotenuse. On a 24 foot span at 6/12, with no ridge deduction, the line is 13.416 feet, which is 13 ft 5 in to the nearest eighth.
Does the overhang add to the line length?
The line length stops at the outside of the wall. The tail is the horizontal overhang divided by the cosine of the pitch angle, and the rafter you cut is the line plus the tail. A 12 inch overhang on a 6/12 is longer than 12 inches, because the rafter climbs.
What is the seat cut?
The pitch angle is arctan(rise / 12). The level cut, the seat of the birdsmouth, is 90 degrees minus that angle. The birdsmouth does not change the line length. The line is still measured along the top edge, from the ridge plumb cut to the wall plumb line.
Is this a rafter length calculator for hips and valleys?
No. This is the common rafter, the one that runs square from plate to ridge. Hips and valleys are a longer triangle. Use this length for the commons only.
Sources
- Common-rafter right triangle. Run = span/2 − half the ridge thickness. Line length = √(run² + rise²). Willis H. Wagner, Modern Carpentry; the same geometry the IRC span tables assume.