Roof pitch and rafters
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Pitch as a ratio and as an angle, common rafter length and roof area from footprint — laying out and covering a roof.
Roof Pitch to Slope Angle
Converts a roof pitch quoted the framing way, p inches of rise per 12 inches of level run, into the slope angle from horizontal, and back.
Common Rafter Length from Run and Pitch
The line length of a common rafter: the level run from ridge to wall plate, stretched by the slope factor for a pitch of p in 12.
Roof Area from Footprint and Pitch
Turns the plan area a roof covers into the sloping area you actually have to shingle, using the pitch multiplier for a pitch of p in 12.
How they fit together
North American roofs are described by a ratio rather than an angle: a 4:12 pitch rises four units for every twelve of horizontal run. The conversion to degrees matters the moment you pick up a speed square or a saw with an angle scale, and 4:12 turns out to be 18.43 degrees.
Rafter length is the hypotenuse over the run, before any allowance for overhang, ridge thickness or bird's-mouth. Roof area from footprint is the estimator's formula: the plan area multiplied by a slope factor, since a pitched roof always carries more surface than the ground it covers. That difference is exactly why shingle orders based on floor area come up short.