Hip and Valley Rafter Calculator

A hip rafter runs from the ridge to an external corner and a valley rafter to an internal corner; when the intersecting planes share a pitch, both solve identically. In plan the member lies on the 45-degree diagonal of a square whose sides are the common run, so its horizontal run is the common run times the square root of 2, about 1.4142. Its length is the hypotenuse of that diagonal run and the common rise, and because the run grew while the rise did not, the hip is always shallower: a 6/12 common roof gives a hip pitch near 4.24 in 12. Carpenters express this as the 17 rule, since 12 inches of common run corresponds to 16.97 inches of hip run, which is read as 17 on the framing square. Three secondary corrections matter on site. Jack rafters that die into the hip shorten by a constant increment, equal to the spacing times the pitch multiplier, so 16-inch on-center spacing at 6/12 gives about 17.9 inches of difference per jack, and each jack needs a compound cheek cut where it meets the diagonal. The hip itself needs a backing bevel along both top edges, or an equivalent drop at the seat cut, so the sheathing lands flat; that angle is roughly 14.5 degrees at 6/12 and steepens with pitch. Every theoretical length is measured to intersecting centerlines, so the hip is also shortened at the ridge by half the ridge thickness taken on the 45-degree diagonal. All of this assumes equal pitches and a true square corner; unequal-pitch hips shift off 45 degrees and require separate treatment.

Hip roofs are ancient, and the diagonal member has always been the hardest part of a roof to lay out because it demands compound angles rather than the simple plumb and level cuts of a common rafter. Traditional timber framers worked the problem geometrically rather than trigonometrically, developing full-scale layout methods on a scribing floor that generated the true lengths and bevels without any calculation at all. In North America the work moved onto the steel framing square. Squares manufactured from the nineteenth century onward carried rafter tables stamped on the blade, and the second line of those tables gave the length of hip or valley per foot of common run, which is where the memorized 18.0 for a 12/12 roof and 18.0-class values for other pitches come from. Frederick T. Hodgson's books on the steel square, published and repeatedly reprinted in the late nineteenth century, popularized stepping the square along the stock to lay out hips, jacks, and their cheek cuts, and the method was still taught in twentieth-century trade texts and in production framing guides such as Larry Haun's writing on efficient carpentry. Pocket construction calculators in the 1980s, and later software and prefabricated trusses, largely replaced hand layout on tract housing, but the underlying relations - the square root of 2 on the run, the constant jack difference, and the backing bevel - are unchanged and are still what a framer checks a machine-produced cut list against.