Rafter framing is geometry made physical. Every cut (plumb, seat, tail) is determined by the roof pitch, span, and overhang dimensions. A framing square and the unit-run/unit-rise system have been the carpenter's tools for solving this geometry for over a century, and the underlying math is still the fastest way to calculate rafter lengths, determine angles, and lay out bird's mouth cuts accurately.
This guide covers common, hip, and valley rafter geometry, the bird's mouth joint that transfers roof loads to the wall plate, and the overhang calculations that determine fascia and soffit dimensions. Whether you are cutting rafters by hand or programming a saw, the relationships are the same.
Unit Run and Unit Rise
Roof pitch is expressed as inches of rise per 12 inches of horizontal run. A 6/12 pitch rises 6 inches for every 12 inches of run. The unit run is always 12 inches for a common rafter. The unit rise is the pitch number (4, 6, 8, etc.). The unit rafter length (the hypotenuse) is the square root of (12² + rise²). For a 6/12 pitch: sqrt(144 + 36) = sqrt(180) = 13.416 inches per foot of run.
Total rafter length equals the unit rafter length times the number of feet of run. For a 24-foot wide building with a centered ridge, each common rafter has a run of 12 feet. At 6/12 pitch, the rafter length is 13.416 × 12 = 161 inches, or 13 feet 5 inches from the ridge plumb cut to the building line (heel of the bird's mouth). Add the overhang length separately. It uses the same pitch multiplier applied to the horizontal overhang distance.
Rafter Length Calculator
Calculate common, hip, and valley rafter lengths with bird's mouth cuts and roof area.
Bird's Mouth Layout
The bird's mouth is the notch cut where the rafter sits on the wall top plate. It consists of a horizontal seat cut (bearing on the plate) and a vertical plumb cut (against the inside face of the wall). The seat cut should be at least the full width of the top plate, typically 3.5 inches for a 2×4 wall. The plumb cut depth (HAP, or height above plate) must leave enough rafter material above the seat to resist the horizontal thrust from roof loads.
A common rule is to limit the bird's mouth depth to no more than one-third of the rafter depth. For a 2×8 rafter (7.25 inches actual), the maximum seat cut depth is about 2.4 inches, leaving 4.85 inches of HAP. If the seat cut needs to be wider to fully bear on a 2×6 wall (5.5 inches), the rafter depth may need to increase to maintain adequate HAP. IRC Table R802.5.1 lists maximum rafter spans based on species, grade, spacing, and snow load. These assume standard bird's mouth proportions.
Hip and Valley Rafters
Hip and valley rafters run diagonally from the ridge to the plate corner (hip) or from the plate intersection to the ridge (valley). Because they run at 45 degrees in plan view, their unit run is not 12 inches but rather the diagonal of a 12-inch square: sqrt(12² + 12²) = 16.97 inches. The unit hip/valley rafter length is sqrt(16.97² + rise²). For a 6/12 pitch: sqrt(288 + 36) = sqrt(324) = 18 inches per foot of common run.
Hip and valley rafters also need compound angle cuts (cheek cuts) where they meet the ridge or other rafters. The backing angle or drop must be applied so the top edges of the hip rafter align with the jack rafter planes on each side. Jack rafters are shortened commons that run from the plate to the hip rafter, decreasing in length by a constant amount equal to the unit rafter length times the jack spacing divided by 12.
Rafter Length Calculator
Calculate common, hip, and valley rafter lengths with bird's mouth cuts and roof area.
Overhang and Tail Cuts
The rafter tail extends beyond the building line to form the eave overhang. Its length is calculated the same way as the main rafter: overhang run times the pitch multiplier. A 16-inch horizontal overhang at 6/12 pitch has a rafter tail length of (16/12) × 13.416 = 17.9 inches measured along the rafter. The tail cut at the end can be plumb (for a vertical fascia), square (perpendicular to the rafter), or a combination.
Fascia board thickness must be accounted for when calculating the plumb cut location. The rafter tail is typically shortened by the fascia thickness so the fascia face lands at the correct line. For a level soffit (horizontal), the subfascia or lookout height determines the soffit plane. Mark and cut a pattern rafter first, test-fit it, verify the ridge height, plate bearing, overhang projection, and fascia alignment, then use it as a template for the remaining rafters.