SettingsSettings for this calculationUS
The horizontal run of a common rafter on this roof — typically half the building width.
The common rafter's run, not the hip's. On a standard hip the hip travels at 45° in plan, so its own run is this figure times the square root of two — the calculation does that conversion, and entering the hip's diagonal run here would apply it twice. Measure from the centre of the ridge to the outside of the plate for a symmetrical roof; on an unequal-pitch or unequal-span roof, this arrangement does not apply and the hip is not at 45°.
The roof's slope, expressed as inches of rise per 12 inches of horizontal run.
The COMMON rafter's pitch — the slope of the main roof planes — not the hip's. The hip is flatter than the roof it sits in, because it climbs the same rise over a run that is 1.414 times longer, and that is the whole reason a hip needs its own cut list. A 6/12 roof has a hip at roughly 4.24/12, and a hip cut to 6/12 will not meet the ridge or the plate.
The hip or valley timber's thickness, across its narrow face.
Thickness, not depth — the dimension you would measure across the top edge, 45 mm on a 45 x 195 or 1.5 in on a 2x8. It is used for one thing: a hip left at full thickness stands proud of both roof planes, because its top corners sit outside the line where those planes meet. The backing bevel and the drop below are the two ways of dealing with that, and both scale directly with this figure. The member's DEPTH does not enter the angles at all and is not asked for.
The centre-to-centre spacing of the jack rafters framing into the hip or valley.
Measured in PLAN along the wall plate, the same way rafter spacing is always quoted — 400 mm or 600 mm where timber is sold in millimetres, 16 in or 24 in where it is not. Jacks framing into an equal-pitch hip shorten by the same amount every time, and that amount is this spacing multiplied by the common rafter's length per unit of run. Enter the spacing you are actually setting out at: the common difference is proportional to it, so a 600 mm spacing shortens the jacks half as often but by half as much again.
Hip/valley rafter length
19.5 ft
- Plumb cut angle, which is the hip's own inclination
- 19.47 °
- Cheek bevel, each side of the hip's ridge cut
- 43.31 °
- Backing bevel on the hip's top edge
- 18.43 °
- Drop, if the top is left square instead of backed
- 0.03 ft
- Jack rafter cheek bevel
- 41.81 °
- Jack rafters shorten by, each spacing
- 1.49 ft
They open the calculator with your figures already in it
Hip/Valley Rafter Plumb Cut Angular Cutlist: 19.5 ft — shown in imperial, US market. The link sets both, so the result they see is the one on your screen.
Show calculation logicHide calculation logic
How this was calculated
Formula source(s)
- Standard carpentry hip/valley formula: for a 45° plan intersection, hip rafter length = common-rafter run × √(2 + (rise/12)²); plumb cut angle = arctan((rise/12)/√2)
- Compound cuts on an equal-pitch 45° hip: a side cut is arctan(cos θ) taken on that member's own pitch, and the backing bevel is arctan(sin θ) taken on the hip's — both reproduce the rafter tables printed on a framing square, which sets its hip figures out against 17 where the exact diagonal is 12√2 = 16.9706
Inputs used
- Common Rafter Run (Half Building Width)
- 13 ft
- Roof Pitch (rise per 12 run)
- 6
- Hip/Valley Member Thickness
- 1.75 in
- Jack Rafter Spacing (on centre)
- 16 in
Intermediate steps
- Plumb cut angle, which is the hip's own inclination
- 19.47 °
- Cheek bevel, each side of the hip's ridge cut
- 43.31 °
- Backing bevel on the hip's top edge
- 18.43 °
- Drop, if the top is left square instead of backed
- 0.03 ft
- Jack rafter cheek bevel
- 41.81 °
- Jack rafters shorten by, each spacing
- 1.49 ft
What this calculation does not cover
- The figure is the theoretical measuring-line length, from the outside corner of the wall plates to the point where the ridge centrelines cross. It deducts nothing for the ridge or the king common rafter the hip butts into, which is half that member's thickness taken on the 45° diagonal, roughly 32 mm off a 45 mm ridge. It also adds no eaves tail, and a hip tail is longer than a common one: a 400 mm horizontal overhang runs about 600 mm along the hip at 6/12, against 450 mm along a common rafter.
- The plumb cut is one angle of a compound one. It is the hip's inclination, arctan((rise/12)/√2) — 19.5° at 6/12 — and it is the setting off a square crosscut; measured against the rafter's top edge the same cut reads as its complement, 70.5°. The cheek bevels are given above and are 45° in PLAN, which is not the angle you set on a saw: the blade figure is arctan(cos θ) taken on the hip's own inclination, 43.3° at 6/12 against the framing square's 43.4°, because the square sets its hip figures out against 17 where the true diagonal is 16.97.
- The backing bevel and the drop above are alternatives, not a sequence: back the top edge to the two roof planes, or leave it square and set the member lower by the drop. Both come from the member thickness entered and nothing else, so a hip of a different thickness needs the figure again. The SEAT is still not produced — it is a compound cut over the corner of two plates rather than the single plumb-and-level birdsmouth of a common rafter, and its geometry depends on how the plates are lapped at that corner, which this page does not ask about.
- The jack figures above are the common difference and the cheek bevel, which is what a set of jacks needs and is not a cut list of them: no jack is counted, none is named, and the first jack's own length depends on where the first spacing falls from the corner, which is a setting-out decision this page does not make. The difference assumes every jack meets the hip at the same 45° in plan and that the spacing is constant — step the spacing anywhere along the run and the arithmetic restarts from that jack.
- No part of this is a structural check. A hip carries the jacks framing into it and a valley collects load from both roof planes, often with a concentrated load where it picks up the minor ridge, so these members are routinely deeper or doubled compared with the commons beside them. The plan run is √2 longer as well, a 4 m (13 ft) common run becoming a 5.66 m (19 ft) diagonal, so common rafter span tables do not apply.
Estimated cost — your price
This site holds no price list for this material — local prices vary too much to publish honestly. Enter your supplier's price and the result is costed with it.
Computed in your browser — nothing you enter is uploaded. Presented in US customary units and US trade terminology. Where a formula follows a published standard, that standard and its edition are cited beside it on this page; where none governs, the page says so. Local amendments override model codes — verify against the code in force where you build.
Sources checked 2026-09-02 · in the site-wide review of 2026-09-06 · v1.0.1
Regulatory standards & verification citations2
- Standard carpentry hip/valley formula: for a 45° plan intersection, hip rafter length = common-rafter run × √(2 + (rise/12)²); plumb cut angle = arctan((rise/12)/√2)
- Compound cuts on an equal-pitch 45° hip: a side cut is arctan(cos θ) taken on that member's own pitch, and the backing bevel is arctan(sin θ) taken on the hip's — both reproduce the rafter tables printed on a framing square, which sets its hip figures out against 17 where the exact diagonal is 12√2 = 16.9706
Cite this page
Your workspace
Most jobs need more than one number. Add the calculators you need next and they open right here, underneath this one — your figures stay on screen and nothing is lost to a page change.
Tools and safety for this job
To cut framing lumber. Generic types, no brands, no prices.
Protection this work requires
- Hardwood dust is a recognised carcinogen: extract at the tool and wear a P2/N95 respirator when cutting or sanding indoors.
- Saws, grinders and breakers run above 85 dB, where hearing damage accumulates and does not recover: defenders or plugs for every cut, not just the long ones.
Show the 5 tools this job needsHide tools
Essential
Circular saw
Combination or framing square
Tape measure
Recommended
Mitre saw
Optional
Hand saw
Also needed as materials: circular saw blades, mitre saw blades.
what each carpentry tool is for, and the spec that decides which to buy where one does.