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Pitch to degrees to square footage, in one step.
Roof pitch is quoted as rise over a 12-inch run, but the number you actually need is usually something else: degrees for the saw, percent slope for drainage, or the slope factor that converts a footprint into the real surface area you buy shingles for. This converter does all of it — arctan for the angle, √(1 + (pitch/12)²) for the area multiplier, and √(2 + (pitch/12)²) for hips and valleys — plus the code minimums that decide what material is even allowed.
Roofers use it to stop converting pitch in their head: the slope factor out of this is what turns a measured footprint into the squares you actually order, and the hip factor is why valleys eat more material than the plan suggests. Homeowners use it to check a bid — footprint times the multiplier is the area, and anything padded past it needs an explanation.
arctan(6 ÷ 12) = 26.57°. Its slope factor is √1.25 = 1.118, so a 2,000 sq ft footprint under a straight gable at 6/12 carries 2,236 sq ft of actual roof surface — the 11.8% difference is why "squares by footprint" estimates come up short.
Per IRC R905.2, asphalt shingles require at least 2/12, and slopes between 2/12 and 4/12 require double-coverage underlayment. From 4/12 up, standard application applies. Below 2/12 you need a membrane or a low-slope-rated metal system.
A hip or valley rafter travels the diagonal of the roof plan, so its run is the hypotenuse of the common run: the multiplier is √(2 + (pitch/12)²) instead of √(1 + (pitch/12)²). At 6/12 that is 1.500 vs 1.118 — a 20 ft common run becomes a 30 ft hip before the pitch even steepens it further.