Rafter length from run and pitch

Either, because the two cuts are parallel and the distance between two parallel lines is the same wherever you measure it. That is only true once both ends are plumb cuts, which is the normal case. It stops being true the moment one end is square-cut or the tail is cut level for a soffit return, and then the top edge and the bottom edge differ by the board depth multiplied by the pitch — 4 5/8 in on a 2x10 at 6:12.

Because it climbs the same rise across a longer run. On a square corner with equal pitches, the hip travels the diagonal of the plan, which is 16.97 in for every 12 in the common travels. The rise at the top is identical, so the hip's own slope is the common pitch divided by the square root of two: a 6:12 roof carries a hip at 4.24:12. That is why hip layout swaps the 12 on the framing square blade for a 17 — and why the seat cut on a hip is a different angle from the seat cut on the commons beside it.

They are this calculation, precomputed for each whole pitch, and they have been stamped on the blade for well over a century. The first line is the length of a common rafter per foot of run, the second is the hip or valley per foot of common run, and the two below are the difference in length of jacks at 16 and 24 in centers. A square gives you four figures per pitch and nothing between them; this page gives the same four for a 6 1/2:12 and for a run of 14 ft 3 1/2 in.

Yes — switch the slope input to total rise and enter the height gained from the top of the wall plate to the underside of the ridge. The pitch falls out of that and the run, and the page shows it so you can sanity-check it against a number anyone frames to. Watch what the drawing measures from: a ridge height taken from finished floor or from grade includes the wall height and will report a roof roughly twice as steep as the one being built.

On an equal-pitch hip with evenly spaced jacks, yes — the common difference is the spacing multiplied by the per-foot factor, so at 16 in centers on a 6:12 every jack is 17.89 in shorter than the one before it. It stops holding on an irregular hip, where the two roof planes have different pitches and the hip does not run at 45 degrees in plan; there the two sides have separate common differences and separate cheek-cut bevels. Uneven spacing at an opening breaks it too, which is why the ladder here lists each jack with its own run.

The headline figure does not — it is the line length from the ridge centerline to the outside of the plate, which is the dimension a plan and a lumber list both use. Enter an overhang and the tail is added below it, on the slope, so you can see how much longer the board is than the projection. The half-ridge shortening is deliberately left out here and handled where the rest of the cuts are, because it changes with the ridge material rather than with the roof.