Measuring & pitch
Rafter maths: run, rise, line length and the birdsmouth
A rafter is two right triangles and a notch. Once the triangles are clear, every cut follows, and the only judgement left is structural.
The parts of a rafter
A common rafter runs from the ridge to the wall plate and past it to form the overhang. Along it there are three cuts and one notch:
- the plumb cut at the top, vertical when the rafter is in place, which sits against the ridge board;
- the birdsmouth, a notch that sits on the wall plate, made of a horizontal seat cut and a vertical heel cut;
- the tail cut at the end of the overhang, often plumb to take the fascia.
Between the plumb cut and the outside of the wall is the line length, the rafter’s length along its top edge from the centre of the ridge to the building line. Everything else is added to or taken from it.
Run, rise and line length
The square root is the pitch factor, the same number that turns a footprint into a roof area. On a 24-foot span at 6/12 the run is 12 feet, the rise 6 feet, and the line length 12 × 1.118 = 13.42 feet, 13 feet 5 inches.
Two adjustments people forget
The ridge board. The run is measured to the centre of the ridge, but the rafter stops at the face of the ridge board, half its thickness short. That half-thickness is horizontal, so along the rafter it becomes half the thickness times the pitch factor: with a 1½-inch ridge at 6/12, 0.84 inches. Shorten every rafter by it, or the ridge rises and the tails come up short.
The overhang. A 12-inch overhang is measured horizontally, but the tail runs along the slope, so it adds 12 × 1.118 = 13.4 inches of rafter, not 12.
A 24-foot span at 6/12 in 2×8
Span 24 ft, 1½ in ridge board, 12 in overhang, 2×8 rafters on a 2×4 wall.
- Line length: 12 × 1.118 = 13.42 ft (13 ft 5 in)
- Shortened for the ridge: (12 − 0.0625) × 1.118 = 13.35 ft (13 ft 4 in)
- Tail: 1 × 1.118 = 1.12 ft
- Total: 14.46 ft, 14 ft 6 in, from a 16-ft board
The rafter length calculator gives the same figures with the cut angles.
The cuts
The plumb cut is made at the pitch angle measured from square: 26.6 degrees at 6/12. The seat cut is its complement, 63.4 degrees. On a framing square both are laid out by putting the rise (6) on the tongue and 12 on the blade, which is why carpenters rarely talk about degrees.
The height above plate (HAP) is the vertical distance from the top of the wall plate to the top of the rafter at the birdsmouth. For a 2×8 (7¼ inches deep) at 6/12 with a 3½-inch seat:
Every rafter on the roof must have the same HAP or the plane will not be flat. The HAP also sets the ridge height: rise plus HAP plus wall height, about 14 feet 6 inches above the floor for this roof, which the ridge height calculator computes.
The birdsmouth limit
Cutting a notch weakens a rafter where it bears on the wall. The International Residential Code (R802.7.1.1) limits notches at the ends of rafters to one quarter of the depth. The depth of material removed, measured square to the rafter, is the seat length times the sine of the angle:
| Rafter | Seat | Pitch | Notch depth | Limit (¼ depth) | Result |
|---|---|---|---|---|---|
| 2×8 (7¼ in) | 3½ in | 6/12 | 1.57 in | 1.81 in | within |
| 2×6 (5½ in) | 3½ in | 12/12 | 2.47 in | 1.38 in | over |
| 2×6 (5½ in) | 5½ in | 12/12 | 3.89 in | 1.38 in | over |
| 2×10 (9¼ in) | 3½ in | 12/12 | 2.47 in | 2.31 in | just over |
Steep pitches with wide plates fail the check fastest. The fix is a deeper rafter, a shorter seat that does not cover the whole plate, or a different bearing detail from the designer.
Hips, valleys and jacks
Hip and valley rafters run diagonally, so their run is the common run times √2 and their length per foot of common run is the hip factor, 1.5 at 6/12. Jack rafters step down along a hip, each shorter than the last by the spacing times the pitch factor: 26.8 inches at 24-inch centres on a 6/12 roof. The hip and valley rafter calculator works these out, with the drop that keeps the sheathing flat over a hip.
Trusses instead of rafters
Most new houses are framed with factory trusses, designed and stamped by the truss maker, rather than cut rafters. The geometry is the same; the cutting and the structural design are done for you, and the job on site is counting and bracing them. The truss count calculator does the counting.
What the geometry does not tell you
None of this says whether a 2×8 can span 12 feet. That depends on the spacing, the lumber species and grade, the dead load of the roofing and the snow load, and it is answered by the span tables in the code (IRC R802.4) or by an engineer. A heavier roof changes the answer; tile and slate weight explains why. Roof pitch explained covers the triangle behind all of it.