Understanding your result
The headline is the hip or valley rafter’s total length from the long point to the tail, with the factor, the plumb cut, the cheek cuts and the board to buy. The stats give the line length, the diagonal run, the plumb cut angle and the jack rafter common difference. The breakdown goes cut by cut, including the shortening, the drop or backing and the jack difference at your spacing.
The number to remember is the line length: common run times the hip factor, from the centre of the ridge to the outside corner of the plates. It is the length you lay out before shortening and before adding the tail, and it is the length the ridge cap calculator wants for each hip. The valley flashing calculator wants the same figure for each valley, plus a foot of run-out.
The common rafters that frame the rest of the roof come from the rafter length calculator, and the pitch factor calculator lists both factors for every pitch.
How we calculate this
The hip factor comes from the hip’s own triangle. Over one foot of common run the hip travels √2 feet horizontally and rises rise ÷ 12 feet, so its length is √(2 + (rise ÷ 12)²) per foot of common run, which is the same as √(rise² + 288) ÷ 12. Because the hip’s run is longer for the same rise, its angle is flatter: a 6/12 roof has a hip at 6 in 17, 19.5 degrees instead of 26.6.
The shortening is different from a common rafter’s because the hip meets the ridge at 45 degrees in plan: it stops short of the ridge centre by half the ridge thickness measured along the diagonal, which is half the thickness times √2. The drop is the amount the hip’s top corner stands above the roof plane when the hip is set to the same HAP as the commons; dropping the hip by that amount (or backing its top edges by the same amount) brings the sheathing to bear. The guide to rafter maths draws the hip’s two triangles and the cheek cuts.
The assumptions behind the numbers
| Assumption | Default | Where it comes from |
|---|---|---|
| Regular hip | 45° in plan | Two planes of equal pitch meeting at a square corner; irregular hips need a different run and cheek angle |
| Hip run | common run × √2 | Geometry of the plan diagonal |
| Ridge board | 1½ in | 2× lumber; 0 for a hip roof without a ridge (pyramid) |
| Hip thickness | 1½ in | 2× lumber; sets the drop or backing |
| Overhang | 12 in horizontal | Common US eave overhang, measured like the commons |
| Jack spacing | 24 in (600 mm) | Typical rafter spacing; 16 in (400 mm) also offered |
| Stock lengths | 8 to 24 ft | Dimensional lumber as sold |
Assumptions last reviewed October 7, 2026.
The calculator does not handle irregular (bastard) hips where the two planes have different pitches, does not size the member, and does not compute the jack rafters’ individual lengths beyond the common difference; the longest jack is the common rafter length less one difference. The pitch guide explains the “per 17” convention for hip pitch.
Two worked examples
A hip on a 24-foot span at 6/12
Span 24 feet (common run 12), pitch 6/12, 1½-inch ridge, 1½-inch hip, 12-inch overhang, jacks at 24 inches.
- Hip run: 12 × √2 = 16.97 ft
- Hip factor: √(36 + 288) ÷ 12 = 1.500
- Line length: 12 × 1.5 = 18 ft
- Shortening: ½ × 1.5 × √2 = 1.06 in
- Tail: 1 × 1.5 = 1.5 ft (1 ft 6 in)
- Total: 18 − 0.09 + 1.5 = 19 ft 5 in; buy a 20-ft board
- Plumb cut 19.5°, cheeks 45°; drop 0.265 in; jack difference 24 × 1.118 = 26.83 in
The hip is a third longer than the 13-ft 5-in common rafter beside it and needs a 20-foot board against the common’s 16. The first jack is 26.83 inches shorter than the common, the next 53.66, and so on down the hip.
A valley on a 10-foot run at 8/12, no ridge, no overhang
A valley where a wing meets the main roof: common run 10 feet, pitch 8/12, no ridge board at the valley, no overhang.
- Valley run: 14.14 ft
- Valley factor: √(64 + 288) ÷ 12 = 1.5635
- Line length and total: 10 × 1.5635 = 15 ft 8 in; buy a 16-ft board
- Plumb cut 25.2°, cheeks 45°; no drop (valleys are not backed; the top edges are bevelled to the V or left square)
- Jack difference at 24 in: 24 × 1.2019 = 28.84 in
The valley takes jacks from both planes, so it will usually be a 2×10 against 2×8 commons, or doubled. The calculator gives the length; the plans give the size.
Where to find your inputs
Common run. Half the span on a symmetrical roof, from the outside of the wall to the centre of the ridge. For a valley, the run of the common rafters of the smaller roof where it meets the larger.
Pitch. The pitch of the planes on both sides of the hip; if they differ, the hip is irregular and this calculator does not apply.
Ridge thickness. The ridge board the hip butts against; on a pyramid or a hip meeting a king common at the ridge end, the shortening differs, so set 0 and shorten on site.
Hip thickness. The lumber the hip is cut from, for the drop or backing.
Common mistakes
- Using the pitch factor for the hip. The hip factor is larger: 1.5 against 1.118 at 6/12. A hip cut with the common’s factor is a foot short.
- Cutting the hip plumb cut at the common’s angle. The hip’s angle is flatter, 19.5 degrees at 6/12. Lay it out with the rise on the tongue and 17 on the blade.
- Forgetting the double cheek. The hip nests between two commons at the ridge and meets two fascias at the tail; both ends need 45-degree cheeks on both sides.
- Leaving the hip square on top. Without the drop or backing the sheathing bears on the hip’s corners and the plane humps along the hip line.
- Shortening like a common. The hip shortens by half the ridge thickness times √2, not times the pitch factor.
- Sizing by length. A 20-foot hip carrying long jacks may need a strut. That is structure, not geometry.
Questions people ask
- How do I calculate a hip rafter length?
- Multiply the common rafter run by the hip factor, √(rise² + 12² + 12²) ÷ 12, for the line length; shorten it by half the ridge thickness times √2; add the overhang times the hip factor for the tail. For a 12-foot common run at 6/12 the hip factor is 1.5, so the line length is 18 feet, the shortening 1.06 inches for a 1½-inch ridge, and the total with a 1-foot overhang 19 feet 5 inches.
- What is the hip rafter run?
- On a regular hip roof the hip runs diagonally across a square whose sides are the common run, so its run is the common run times √2: 16.97 feet for a 12-foot common run. The hip rises the same amount as the commons over that longer run, which is why hip pitch is written in inches per 17 (really 16.97): a 6/12 roof has a 6/17 hip.
- What are the cuts on a hip rafter?
- A plumb cut at the ridge at the hip's own angle (arctan(rise ÷ 16.97), 19.5 degrees for a 6/12 roof), with the top end also cut at 45 degrees on both sides (a double cheek) so it nests between the two common rafters at the ridge; a birdsmouth at the corner of the plates; and a tail with the same double 45-degree cheek so the fascia meets square. A valley rafter is cut the same way but its cheeks point the other way.
- What is hip drop and backing?
- The top of a hip rafter is a ridge between two roof planes, and if it were left square the sheathing would bear on its corners and sit high. Either the hip is dropped, by cutting its birdsmouth deeper by the drop amount so its corners fall into the planes, or it is backed, by bevelling its top edges. The drop is half the hip's thickness times the rise per 16.97: 0.265 inches for a 1½-inch hip at 6/12.
- What is the jack rafter common difference?
- Jack rafters run from the hip to the plate (or from the ridge to the valley), each one shorter than the last by the same amount: the spacing times the pitch factor. At 24-inch centres on a 6/12 roof every jack is 24 × 1.118 = 26.83 inches shorter than its neighbour. Cut the longest, then subtract the common difference for each one down the hip.
- How big should a valley rafter be?
- That is a structural question the calculator does not answer: it gives the length and the cuts. In practice a valley rafter carries the jacks of two planes and is usually one size deeper than the commons or doubled, and a hip supporting long jacks may need a strut to a bearing wall. The plans, the span tables or an engineer decide.