Hip & Valley Rafter Length Calculator

Why a hip rafter uses 17 where a common rafter uses 12

A hip rafter runs diagonally up the outside corner where two roof planes meet; a valley rafter runs down the inside corner where they meet. On an equal-pitch roof both sit at exactly 45° in plan, and that single fact drives every number on this page.

Walk out one foot of run on a common rafter and you have moved 12 inches horizontally. Walk out the same amount of rise on the hip and you have moved 12√2 = 16.97 inches horizontally, because the hip is travelling along the diagonal of a 12-inch square. Carpenters round that to 17, and it is why the framing square's hip and valley tables are read against 17 on the blade rather than 12.

The consequences follow immediately. The hip is longer than the common rafter serving the same corner, because it covers more horizontal distance for the same rise. It is also flatter: a 6/12 common rafter sits at 26.57° from horizontal, while its hip sits at 19.47°. And because the hip is flatter, its plumb cut is a shallower angle and its cheek cuts are closer to a plain 45° than the jack rafters' are.

A valley rafter obeys the same arithmetic with the sign of the corner reversed. Every length, angle and difference this calculator returns applies unchanged to a valley of the same pitch and run.

The formula, term by term

Start with the right triangle the hip actually lives in. Its horizontal leg is the hip run R√2, its vertical leg is the total rise R·x/12, and the hip itself is the hypotenuse.

Hip length. L = √( (R√2)² + (Rx/12)² ) = R√(2 + (x/12)²). Multiply out by 12 and you get the working form: L = R × √(288 + x²) / 12, where √(288 + x²) is the unit hip length — inches of hip per 12 inches of common run. At 6/12 it is exactly 18.000, which is why a 6/12 hip is one of the few roofs that comes out in whole inches.

Hip slope angle. arctan(x / 16.97). This is the angle from horizontal, and it is also the angle you mark from square when you scribe the plumb cut across the hip's face.

Cheek cut angle. Where the hip dies into the ridge it needs a double cheek cut, each face at 45° in plan. Because the rafter is tilted, a 45° plan angle does not appear as 45° on the timber: the along-the-rafter direction is foreshortened by cosθ when you look at it from above, so the cut you actually mark is A = arctan(1/cosθ). Written with the unit lengths it is arctan(√(288+x²) / 16.97) for the hip and arctan(√(144+x²) / 12) for a jack. Both are always greater than 45°, and both approach 45° only as the pitch approaches flat.

Jack common difference. Jacks are spaced s apart along the plate. Each step of s in run adds s·√(144+x²)/12 of rafter length, and that increment never changes — which is why a stack of jacks can be cut as an arithmetic progression rather than measured one at a time.

Shortening. The theoretical hip runs to the ridge centreline. The real one stops at the ridge face, so you take off half the ridge's 45° thickness measured along the hip. Each jack stops at the hip's cheek, so you take off half the hip's 45° thickness measured along the jack.

Reading the angles onto real timber

The plumb cut angle is measured from square, not from the edge. Set a speed square or bevel gauge to 19.47° for the 6/12 hip above and the line you scribe is vertical when the rafter is in place. If your bevel gauge reads from the blade instead, use the complement, 70.53°. Getting this backwards is the fastest way to cut a rafter twice.

Cheek cuts are always steeper than 45°. That surprises people who expect a 45° corner to give a 45° cut. The reason is foreshortening: the rafter is tilted, so its plan projection is compressed along its length and a 45° plan angle opens out on the timber. At 6/12 the jack cheek is 48.19°; at 12/12 it is 54.74°. Both are outside the 45° detent on a standard mitre saw, so on anything but a shallow roof you mark the top edge and cut with a circular saw.

The hip's cheek is always closer to 45° than the jack's. That is not a rule of thumb, it is the algebra: the hip's angle is arctan(√(288+x²)/16.97) and the jack's is arctan(√(144+x²)/12), and the first is smaller at every positive pitch. It gives you a useful self-check — if your hip cheek comes out steeper than your jack cheek, you have swapped 12 and 17 somewhere.

The common difference is a cutting instruction. Cut the longest jack, then chop the common difference off each successive one. On the 6/12 example that is 17⅞ in per jack, and the whole stack of eight comes off one setup. Verify against the schedule the calculator prints before the crew starts, because an error in the common difference multiplies down the whole stack.

Backing or dropping the hip. The hip's top edge is an arris, not a flat, so you either bevel both top edges (backing) or set the whole member lower in the corner (dropping). Neither changes a length or angle on this page, but both change the seat cut — size that with the birdsmouth and HAP calculator.

Unit lengths and cut angles by pitch

Two pitches land on round numbers and are worth remembering: 5/12 gives a unit common of exactly 13, and 6/12 gives a unit hip of exactly 18.

Mistakes that cost a hip rafter

• Measuring the run to the ridge face instead of the centreline. The theoretical length runs to the centreline; the shortening is then applied once. Do both and you have shortened the rafter twice.
• Using 12 instead of 17 on the framing square. This produces a hip cut at the common rafter's slope. It will not reach the ridge and it will not sit down on the corner.
• Expecting a 45° cheek cut. The plan angle is 45°; the cut on the timber never is. At 8/12 the jack cheek is 50.24°.
• Forgetting the hip's overhang is longer than the common's. A 24 in horizontal overhang becomes 36.00 in of hip on a 6/12 roof — the same √2 stretch as the rest of the run.
• Neglecting to back or drop the hip. Sheathing landing on an unbacked hip rocks on the arris and telegraphs through the roof covering.
• Applying jack shortening in the wrong direction. Jacks lose length at the hip end and keep their full plumb cut at the plate end. The birdsmouth position does not move.
• Cutting the whole stack before checking the first jack in place. Dry-fit the longest jack, confirm the cheek angle against the hip, and only then run the common difference down the pile.

Where hip geometry sits in the framing sequence

Establish the pitch first. If you are working from an existing roof rather than a drawing, take it off with a level and a rule and confirm it with the roof pitch calculator. Everything downstream — unit lengths, angles, differences — is a function of that one number.

Cut the common rafters next, using the rafter length calculator, because the hip has to land at the same ridge height they establish. Then cut the hips, then the jacks in descending order. Seat cuts on all three come from the same birdsmouth geometry, and the height above plate has to match across commons, hips and jacks or the roof plane will not be flat.

Once the roof is framed and sheathed, the sloped area you need for a shingle order comes straight out of the same unit common length: multiply the plan area by √(144+x²)/12 and hand the result to the shingle bundle calculator. For the trim that follows — fascia returns, soffit mitres and any moulding that runs up a hip — the compound angles come from the compound mitre angle calculator, which solves the same foreshortening problem for a different pair of planes.

Two related layouts are worth naming. A Dutch hip stops the hips short of the ridge and finishes with a small gable, so the hips are cut to a shorter run but at the same angles. A cripple jack runs between a hip and a valley and touches neither plate nor ridge; its length is the difference between two positions on the schedule, and its two ends take opposite-hand cheek cuts.

Framing terms used here

Run

The horizontal distance a rafter covers. For a common rafter it is half the building width; for a hip it is that run multiplied by √2.

Unit length

Rafter length per 12 inches of run — √(144 + x²) for a common rafter and √(288 + x²) for a hip or valley on the same roof.

Plumb cut

The vertical cut at the ridge or heel end of a rafter. Marked at the rafter's slope angle measured from square across the face.

Cheek cut

The angled side cut where a hip meets a ridge or a jack meets a hip. Also called a side cut. Hips take two; jacks take one.

Jack rafter

A shortened common rafter running from the plate to a hip, or from a valley to the ridge. Successive jacks differ by a constant length.

Common difference

The fixed length change between adjacent jack rafters: spacing × unit common length ÷ 12.

Backing the hip

Bevelling both top edges of a hip so the sheathing on each roof plane bears flat on it instead of on the arris.