Calculate hip rafter length, backing angle, and side cut angles for hip roof framing. Accurate geometry for complex roof structures.
Enter the roof pitch as rise per 12 inches of common run and the common rafter run in feet. A hip rafter travels diagonally at 45° in plan from the building corner to the ridge, so it covers √2 times more horizontal distance than the common rafters it meets.
Hip length is computed as common run × √(2 + (rise/12)²). With a 12 ft common run at 6/12 pitch the factor is √2.25 = 1.5, giving an 18.0 ft hip — compared with about 13.42 ft for the common rafter on the same roof.
The hip pitch angle is atan(pitch ÷ √2), about 19.5° for a 6/12 roof versus 26.6° for the commons. This is the basis of the carpenter's convention of cutting hips at the same rise over 17 inches of run, since 12 × √2 ≈ 16.97.
It spans the plan diagonal. Per unit of common run the hip length factor is √(2 + (rise/12)²) versus √(1 + (rise/12)²) for a common — at 6/12 that is 1.50 versus about 1.12, which is why a 12 ft run produces an 18 ft hip but only a 13.42 ft common.
The hip climbs the same rise while covering √2 times the horizontal distance, so its angle is atan((rise/12) ÷ √2). A 6/12 roof has 26.6° commons but a 19.5° hip — the reason hip plumb cuts use a different angle setting.
For every 12 inches a common rafter runs, the hip under the same roof surface runs 12√2 ≈ 16.97 inches, rounded to 17 on framing squares. Stepping off a hip with the square set at rise over 17 reproduces the same geometry this calculator solves exactly.
Yes. A regular valley rafter lies on the mirror-image diagonal of a hip — 45° in plan — so its length and slope angle follow identical formulas. Enter the common run that meets the valley and read the results the same way.