Hips: why the pitch is shallower and the length is longer

A hip climbs the same rise over a longer run, which makes it shallower than the roof it sits in and longer than any common.

On a square-cornered hipped roof with the same pitch on all four faces, the hip runs at 45 degrees across the plan. That single fact explains everything else about it.

The plan run

A common rafter's run is the half span. On a 4800 span that is 2400. The hip starts at the same corner and finishes at the same apex, but it travels diagonally to get there, so its run is the diagonal of a square with 2400 sides:

plan run = √(2400² + 2400²) = 3394 mm

Same rise, longer run. Both members climb 1680 to the apex.

The pitch

Divide the same rise over a longer run and you get a shallower angle. At 35 degrees on the commons:

θₕ = atan(tan 35° ÷ √2) = 26.3°

That figure is the hip's own pitch, and it is what the hip's plumb and seat cuts are set from. Marking a hip with the roof pitch is one of the classic ways to lose a stick: the cut looks plausible and the member arrives at the corner at the wrong height.

The seat cut follows from it in the usual way, 63.7 degrees, and the hip's double cheek cut at the ridge is a separate angle again.

The length

The hip's length to the apex is the diagonal run and the rise worked together:

√(3394² + 1680²) = 3787 mm

The hip's cut length works out at 3750 against a common rafter's 2910 on the same roof, over 800mm more timber in the same building, which is why hips are usually ordered a section deeper than the commons. Here they are 47 by 172 against 47 by 147 for the rafters.

What comes off it

The 3787 is the length to the theoretical apex, and no hip is ever cut to it. Something is in the way at the top: on this roof, the crown rafter. The hip's centreline meets the side face of that rafter 33mm short of the apex on plan, which is 37mm measured along the hip.

3787 − 37 = 3750 mm

Then the hip's own breadth gives it two ends rather than one. Its double cheek cut sets the short point back a further 26mm from the long point, so the schedule prints 3750 to the long point and 3724 to the short point. Both numbers matter at the bench, because you mark one and cut to the other.

The one to remember

Four hips on a standard hipped roof, all identical on a square plan with equal pitch. Cut one, check it in place, then use it as the pattern. Getting a hip wrong is expensive in a way that getting a jack wrong is not.

Every figure above is worked out by the same engine that produces CutRoof's cut schedules, and is checked against it on every build.