Common Rafter Length from Run and Pitch

A rafter is the hypotenuse of a triangle whose base is the level run, half the span, and whose height is the rise. Factor the run out of Pythagoras and you get $L = R\sqrt{1 + (p/12)^2}$, where the square root is the number stamped on the framing square as the "length of common rafter per foot of run". For a 5:12 roof that factor is exactly 13/12, since 5-12-13 is a Pythagorean triple, so a 12 ft run gives a 13 ft rafter with no rounding at all.

What you get is the line length, measured from the centre of the ridge to the outside of the wall plate. Real rafters are longer at one end and shorter at the other: add the overhang or tail you want, then subtract half the ridge board thickness along the slope. Miss that second step and every rafter in the roof is long by the same amount, which shows up as a ridge that will not close.

The 9:12 pitch is worth memorising because its factor is a clean 1.25, the 3-4-5 triangle wearing a roofer's hat.