What the birdsmouth does and why HAP matters
A rafter arrives at the wall on a slope and has to bear on a plate that is dead level. The birdsmouth is the L-shaped notch that reconciles the two: a level seat cut that sits on the plate, and a plumb heel cut that drops from the outer end of the seat to the bottom edge of the rafter. Cut it right and the rafter transfers load through the full width of the plate. Cut it too deep and you have removed section from the member exactly where the shear is highest.
Height above plate — HAP, sometimes called heel stand or heel height — is the plumb distance from the top of the plate to the top edge of the rafter at the heel. It is the number that actually controls the finished roof, for three reasons. It sets the ridge height, because the top of the rafter at the ridge is the total rise plus HAP. It sets the fascia and soffit line at the eave. And it has to be identical on every rafter in the roof, including hips, valleys and jacks, or the roof plane will not be flat.
HAP is also where framers meet the insulation problem. Energy codes want full-depth insulation over the top plate, and at a shallow pitch a conventional birdsmouth leaves nowhere near enough room. That is why raised-heel (energy) trusses exist, and why on a stick-framed roof you sometimes give up the notch entirely and hang the rafter off a sloped seat plate instead.
The geometry, term by term
Set the top of the plate as your datum and work along a plumb line at the heel.
Plumb depth: D / cos θ. A rafter of actual depth D measured square to itself is deeper than D when you measure it vertically, because it is tilted. A 2×8 is 7.25 in deep square, but 8.106 in deep plumb at 6/12. This is the same foreshortening that makes every plumb cut on a rafter longer than the rafter is wide.
Heel drop: S · tan θ. The seat and the rafter's bottom edge intersect at the inner end of the seat. Travel back out along the seat by its length S and the bottom edge has fallen below the seat plane by S·tan θ. That drop is the height of the plumb heel cut.
HAP = D / cos θ − S · tan θ. Total plumb depth, less the part of it that the notch has taken away.
Notch depth: S · sin θ. The code limit is written square to the member, not plumb, so the heel height is not the number to check. Multiply the heel height by cos θ — equivalently, multiply the seat length by sin θ — and you get the perpendicular depth that actually has to clear the limit. At 6/12 the difference matters: 1.750 in of heel is only 1.565 in of notch.
Remaining depth: D − S · sin θ. What is left of the rafter above the seat, square to the member. This is the section that carries the load across the bearing.
One consequence is worth stating in advance because it catches people out. HAP is not a one-way function of pitch. Raising the pitch increases the plumb depth but also increases the heel drop, and which effect wins depends on the seat length. Differentiate and the turning point falls where D·sin θ = S, at which HAP equals exactly √(D² − S²). Below that pitch HAP falls as pitch rises; above it, HAP rises.
Worked example: a 2×8 rafter at 6/12 on a 2×4 wall
A 2×8 rafter — 7.25 in actual depth — at 6/12 pitch, seated on the full 3.5 in width of a 2×4 top plate, with a 12 ft run to the ridge.
- Slope angle. θ = arctan(6 ÷ 12) = 26.565°, so cos θ = 0.894427, sin θ = 0.447214 and tan θ = 0.500000.
- Plumb depth. 7.25 ÷ 0.894427 = 8.1057 in.
- Heel cut height. 3.5 × 0.500000 = 1.750 in. That is the plumb leg of the notch.
- Height above plate. 8.1057 − 1.750 = 6.356 in, near enough 6⅜ in. Mark this on every rafter, hip and jack in the roof.
- Notch depth square to the rafter. 3.5 × 0.447214 = 1.565 in.
- Code limit. 7.25 ÷ 4 = 1.8125 in. The notch passes with 0.247 in to spare.
- Remaining depth. 7.25 − 1.565 = 5.685 in of rafter over the bearing.
- Ridge height. Rise over a 12 ft run at 6/12 is 12 × 6 = 72 in, so the top of the rafter at the ridge centreline sits 72 + 6.356 = 78.356 in above the plate. That is the number your ridge board's top edge has to reach.
Now change one thing. Keep the 2×8 and the 6/12 pitch but move to a 2×6 wall, so the seat becomes 5.5 in. The heel drops to 5.5 × 0.5 = 2.750 in, HAP falls to 8.1057 − 2.750 = 5.356 in, and the notch deepens to 5.5 × 0.447214 = 2.460 in — well past the 1.8125 in limit. On a 2×6 wall at this pitch a 2×8 rafter cannot take a full-width seat. Either step up to a 2×10, or seat the rafter on part of the plate width and accept a bearing check.
Reading the result
The notch check is the first thing to look at. The 2021 International Residential Code limits a notch at the end of a sawn rafter or ceiling joist to one-quarter of the member's depth. That is a hard number for stick framing and inspectors do check it. If your notch fails, you have three honest ways out: a deeper rafter, a shorter seat cut, or no notch at all — bearing the rafter on a bevelled plate or a sloped-seat hanger.
Bearing is the second. A rafter needs at least 1½ in of bearing on wood. Shortening the seat to pass the notch check quickly runs into that floor, and a seat under 1½ in is not a solution, it is a different violation.
HAP is a decision, not just an output. Framers commonly pick a round HAP — 5½ in, 6 in, 7¼ in — and cut the seat to suit rather than the other way round. Set a target HAP in the advanced group and this calculator solves the seat length backwards. Two reasons to do that: matching an existing roof you are tying into, and clearing enough space over the plate for full-depth insulation without a raised heel.
Watch what happens at low pitch. On a 3/12 or 4/12 roof, a 2×8 on a 2×4 wall gives 6.598 in and 6.475 in of HAP respectively, which sounds generous until you subtract the sheathing and remember that the insulation has to taper down to nothing at the fascia. The chart and the pitch schedule this calculator prints trace the whole curve for your own inputs, including the point where it turns.
Hips, valleys and jacks share the HAP, not the seat. A hip rafter sits at a flatter angle than the commons on the same roof, so its seat cut geometry is different even though its HAP must match. Work the hip's own slope out with the hip and valley rafter calculator and re-run this page with that angle.
Height above plate and notch depth for common rafter sizes at 6/12
| Rafter | Actual depth (in) | Plate | Seat (in) | HAP (in) | Notch (in) | Limit (in) | Within limit |
|---|---|---|---|---|---|---|---|
| 2×6 | 5.50 | 2×4 | 3.50 | 4.399 | 1.565 | 1.375 | No |
| 2×8 | 7.25 | 2×4 | 3.50 | 6.356 | 1.565 | 1.813 | Yes |
| 2×10 | 9.25 | 2×4 | 3.50 | 8.592 | 1.565 | 2.313 | Yes |
| 2×12 | 11.25 | 2×4 | 3.50 | 10.828 | 1.565 | 2.813 | Yes |
| 2×6 | 5.50 | 2×6 | 5.50 | 3.399 | 2.460 | 1.375 | No |
| 2×8 | 7.25 | 2×6 | 5.50 | 5.356 | 2.460 | 1.813 | No |
| 2×10 | 9.25 | 2×6 | 5.50 | 7.592 | 2.460 | 2.313 | No |
| 2×12 | 11.25 | 2×6 | 5.50 | 9.828 | 2.460 | 2.813 | Yes |
Notch depth depends only on the seat length and the pitch, so it is identical down each block. What changes is the limit, which is a quarter of the rafter's own depth — which is why the same seat passes on a 2×10 and fails on a 2×6.
Never notch an engineered rafter
The quarter-depth rule is written for sawn lumber. Wood I-joists, LVL, LSL and parallel-strand rafters carry their bending stress in thin, highly stressed flanges, and a birdsmouth cut into a flange removes the part of the section that is doing the work. Manufacturers prohibit it outright, with the single common exception of a shallow seat cut into the bottom flange of an I-joist at the bearing, and only where their literature explicitly permits it.
The alternatives are a bevelled bearing plate, a sloped-seat hanger, or a structural ridge with the rafters hung rather than seated. All three keep the full section over the bearing, and all three change your HAP, so recompute the ridge height before you cut anything.
Mistakes that show up after the sheathing goes on
- Checking the heel height against the notch limit. The limit is measured square to the rafter, and the heel height is measured plumb. At 6/12 a 1.750 in heel is a 1.565 in notch — an 11% difference, and it grows with pitch.
- Using nominal depth instead of actual. A 2×8 is 7.25 in, not 8 in. Use 8 and your quarter-depth limit is 2.000 in instead of 1.813 in, which is exactly the sort of error that passes your own check and fails the inspector's.
- Letting the HAP vary between rafters. Commons, hips, valleys and jacks all have to arrive at the same height above the plate. If they do not, the roof plane rocks and the sheathing telegraphs every discrepancy.
- Cutting the seat past the outside face of the wall. The heel plumb cut belongs at the outer face of the plate. A seat longer than the plate leaves the heel cantilevered over nothing.
- Forgetting that HAP sets the ridge height. Ridge top = rise + HAP. Cut the ridge to the rise alone and every rafter will stand proud of it.
- Overcutting the notch corner with a circular saw. Running the blade past the intersection cuts a slot into the remaining section. Stop short and finish the corner with a handsaw or a jigsaw.
- Assuming a steeper roof always gives a smaller HAP. It does below the turning point at D·sin θ = S, and it does the opposite above it.
Alternatives to notching, and where this fits
The birdsmouth is the default for stick framing, but it is not the only detail. A raised-heel truss lifts the top chord above the plate on a vertical block, giving full insulation depth over the wall and removing the notch entirely. A bevelled seat plate — a wedge ripped to the roof angle and nailed to the plate — lets an unnotched rafter bear flat, which is the usual answer for engineered members. A sloped-seat hanger does the same job in steel and is the standard detail where rafters meet a structural ridge or a hip beam.
Before you finalise a rafter size, check the span rather than just the notch: a member that passes the quarter-depth rule can still be undersized for its span and load. Start from the pitch — take it off site with the roof pitch calculator — then get the rafter length from the rafter length calculator, and size the member against snow and dead load with the roof snow load calculator. The same notching arithmetic applies to floor framing, where the equivalent question is answered by the floor joist span calculator.
One historical note that still shapes the detail: the quarter-depth end-notch allowance exists because shear stress in a bending member peaks at the supports and is carried mainly by the section near the neutral axis. Notching from the bottom at a bearing removes material below the neutral axis and creates a re-entrant corner where a split can start. That is why the rule is stricter at the ends than the third-depth workshop convention many carpenters carry in their heads, and why you stop the saw cut short of the corner.
Terms on the cut list
- Seat cut
- The level cut in the birdsmouth that bears on the top plate. Its length is the bearing length.
- Heel cut
- The plumb cut at the outer end of the seat, dropping to the bottom edge of the rafter. Also called the plumb cut of the birdsmouth.
- HAP
- Height above plate: the plumb distance from the top of the plate to the top edge of the rafter at the heel. Also called heel stand.
- Plumb depth
- The rafter's depth measured on a vertical line rather than square to the member: D ÷ cos θ.
- Raised heel
- A framing detail that lifts the rafter or truss top chord above the plate to make room for full-depth insulation over the exterior wall.
- Tail
- The part of the rafter beyond the heel cut that forms the eave overhang. Its length is unaffected by the birdsmouth geometry.