Every cut on a rafter — ridge, birdsmouth, tail, hip, jack cheek — is decided by one number: the pitch angle. This calculator turns any pitch into the full set of angles you need at the saw bench: plumb and seat cuts for common rafters, the shallower plumb and seat angles of hips and valleys, the cheek cut for jack rafters, and the miter saw settings that produce each one. All angles here are measured from the top edge of the board, the way you mark them with a framing or speed square.
| Pitch | Angle | Plumb | Seat | Hip plumb | Hip seat | Cheek |
|---|---|---|---|---|---|---|
| 1 / 12 | 4.8° | 85.2° | 4.8° | 86.6° | 3.4° | 44.9° |
| 2 / 12 | 9.5° | 80.5° | 9.5° | 83.3° | 6.7° | 44.6° |
| 3 / 12 | 14.0° | 76.0° | 14.0° | 80.0° | 10.0° | 44.1° |
| 4 / 12 | 18.4° | 71.6° | 18.4° | 76.7° | 13.3° | 43.5° |
| 5 / 12 | 22.6° | 67.4° | 22.6° | 73.6° | 16.4° | 42.7° |
| 6 / 12 | 26.6° | 63.4° | 26.6° | 70.5° | 19.5° | 41.8° |
| 7 / 12 | 30.3° | 59.7° | 30.3° | 67.6° | 22.4° | 40.8° |
| 8 / 12 | 33.7° | 56.3° | 33.7° | 64.8° | 25.2° | 39.8° |
| 9 / 12 | 36.9° | 53.1° | 36.9° | 62.1° | 27.9° | 38.7° |
| 10 / 12 | 39.8° | 50.2° | 39.8° | 59.5° | 30.5° | 37.5° |
| 11 / 12 | 42.5° | 47.5° | 42.5° | 57.0° | 33.0° | 36.4° |
| 12 / 12 | 45.0° | 45.0° | 45.0° | 54.7° | 35.3° | 35.3° |
The pitch angle θ (theta) is the angle the rafter makes with the horizontal: θ = atan(pitch ÷ 12). A 6/12 pitch gives θ = 26.57°. From that single angle, the common rafter's two working cuts fall out immediately. The plumb cut — the vertical face at the ridge and at the birdsmouth — sits at 90° − θ from the top edge of the board: 63.4° at 6/12. The seat cut — the horizontal face that rests on the plate — sits at θ from the top edge: 26.6° at 6/12. Plumb and seat always add to 90°, because one is vertical and the other horizontal and the board crosses both.
That is the entire system. Hip and valley rafters run at a shallower angle because they travel diagonally across the same rise; jack rafters keep the common pitch but add a sideways cheek cut where they meet the hip. Each of those angles is a variation on θ, which is why a framer who knows the pitch can lay out a whole roof with one square and no tables.
| Pitch | Pitch angle θ | Plumb cut | Seat cut | Hip plumb | Hip seat | Jack cheek |
|---|---|---|---|---|---|---|
| 3/12 | 14.0° | 76.0° | 14.0° | 80.0° | 10.0° | 44.1° |
| 4/12 | 18.4° | 71.6° | 18.4° | 76.7° | 13.3° | 43.5° |
| 6/12 | 26.6° | 63.4° | 26.6° | 70.5° | 19.5° | 41.8° |
| 8/12 | 33.7° | 56.3° | 33.7° | 64.8° | 25.2° | 39.8° |
| 10/12 | 39.8° | 50.2° | 39.8° | 59.5° | 30.5° | 37.5° |
| 12/12 | 45.0° | 45.0° | 45.0° | 54.7° | 35.3° | 35.3° |
A hip rafter climbs the same rise as the common rafter but covers more horizontal ground doing it — it runs diagonally, so its run is the common run times √2. Same rise, longer run, shallower angle: the hip angle is atan(tan θ ÷ √2). At 6/12 the hip climbs at only 19.5° even though the roof planes are at 26.6°. Its plumb cut is therefore 90° − 19.5° = 70.5° from its top edge, and its seat cut 19.5°. Set a hip out with the common rafter's angles and the ridge joint will gape at the top and bind at the bottom.
Valleys obey the identical geometry — a valley is a hip turned inside out — so every hip angle in the table above serves for valleys too, on equal-pitch intersections. Jack rafters keep the common plumb and seat angles, but where a jack meets the hip its end is sliced sideways at the cheek cut: 41.8° from the edge at 6/12, marked across the top face and cut with the saw blade bevelled to match. The cheek angle changes surprisingly little across pitches — it stays between about 44° and 35° from 3/12 to 12/12 — because it is mostly set by the 45° plan angle of the corner.
Here's the trap that ruins first attempts: a miter saw's scale reads from a square crosscut, not from the board's edge. A cut that sits at angle A from the top edge is produced by setting the saw to 90° − A. So for a common rafter the settings are the mirror of the marked angles — the plumb cut is set to θ (26.6° at 6/12) and the seat cut is set to 90° − θ (63.4°). If you set the saw to the number you marked, you'll cut the complement and the birdsmouth will face the wrong way.
On a circular saw with a speed square the marked angle is used directly — the square pivots on the board's edge and reads the angle from it, which is why site framers think in from-the-edge angles and bench framers think in saw settings. This calculator reports both. Whichever tool you use, cut one rafter, offer it up, and check the ridge and plate contact before cutting the batch.
An 8/12 roof has θ = atan(8 ÷ 12) = 33.7°. The cards below are the complete cutting list for that roof — common, hip and jack — all derived from that one angle.
Plumb cut = 90 − 33.7 = 56.3° from the top edge; seat cut = 33.7°. On the miter saw those become settings of 33.7° and 56.3° respectively.
Hip angle = atan(tan 33.7° ÷ √2) = atan(0.471) = 25.2°. Plumb = 90 − 25.2 = 64.8° from the edge; the seat uses the hip angle itself.
The jack keeps the common plumb and seat angles, and its cheek — where it dies into the hip — is cut at 39.8° from the edge, with the blade bevelled to suit the hip's slope.
An angle is only useful at the right position on the board: the plumb cut belongs at the ridge end of the line length, the seat cut where the rafter crosses the wall. Once your angles are set, get the distances they belong at from the rafter length calculator, which works out line length, tail and ridge deduction for the same pitch. If your roof has a birdsmouth, its plumb face uses the plumb angle from this page, and the birdsmouth calculator turns your seat length into the notch depth and HAP that go with it. Angles, lengths and notch from the same pitch will always agree with each other — if a test rafter doesn't sit right, one of the three was taken from a different pitch.
Every result on this page comes from the method described above, using standard framing geometry and the published data credited below. Found a number that looks wrong? Use Report an Error — every report is read.
63.4° measured from the top edge of the board. The pitch angle is atan(6÷12) = 26.57°, and the plumb cut is its complement, 90 − 26.57 = 63.4°. On a miter saw you set 26.6°, because the saw reads from square.
The hip climbs the same rise over a longer, diagonal run (the common run × √2), so its pitch angle is shallower: atan(tan θ ÷ √2). At 6/12 that's 19.5° instead of 26.6°, giving a plumb cut of 70.5° instead of 63.4°.
Both describe the same cut from opposite references. Angles on this page are measured from the board's top edge — use them directly with a framing or speed square. A miter saw reads from a square crosscut, so set 90° minus the marked angle.
The cheek cut is the angled slice on the end of a jack rafter where it meets the hip or valley. It's marked across the top face from the board's edge — 41.8° at 6/12, and it only moves between about 44° and 35° across all common pitches.
Yes, when the two intersecting roofs have equal pitches. A valley is the same geometry as a hip, concave instead of convex, so the plumb, seat and cheek angles are identical. Unequal pitches need the angles worked out for each plane separately.
Yes — that's what a framing square is for. Set the rise figure on the tongue and 12 on the blade and the square's edges give you plumb and seat lines directly. This calculator is the check on that layout, and gives you the hip and cheek angles the square alone doesn't.
The full common rafter: length, angles, birdsmouth and materials.
Line and cut lengths and angles for the diagonal rafter on a hip roof.
Where two roof planes meet — valley length and its cut angles.
First jack length and the common difference for your spacing.
Single-slope rafters between two different wall heights.
Seat cut, heel height and HAP, checked against the IRC one-quarter limit.
Convert pitch, rise/run, degrees and percent, both ways.
How far a board can span — IRC tables by species, grade and load.
The lumber size a span actually needs. The inverse of the span tool.
Two pitches, one roof — upper and lower rafter lengths.
The full pitch × run table, printable for the workshop wall.
Figure any rafter by hand — formula and framing-square methods.