Calculate common, hip, valley, and jack rafter lengths, plumb and seat cut angles, ridge deductions, eave overhang tails, and IRC maximum lumber span limits.
Cutting accurate roof rafters requires mastering the right triangle formed by the rafter run, ridge rise, and hypotenuse line length:
Line length is measured from the long point of the plumb ridge cut to the heel of the birdsmouth. Total board length includes the sloped eave overhang tail plus extra stock for subfascia cuts.
Always subtract half the ridge thickness (0.75" for 2x stock) measured horizontally or 90° from the plumb cut line. Failing to subtract this causes the roof to peak too high and push exterior walls outward.
A timber rafter is shaped by 3 critical angular cuts:
The vertical angle where the rafter abuts the central ridge board ($\theta = \arctan(\text{Pitch}/12)$). Scribed using the roof pitch number on the Swanson speed square common scale.
A level $90^\circ$ horizontal cut resting on the wall plate. Max vertical heel cut depth must never exceed $1/3$ the rafter width (IRC Section R802.7.1) to prevent beam shear failure.
Cut either plumb (vertical for gutter subfascia) or square (perpendicular to rafter slope for exposed decorative rafter tails).
Rafters exert outward horizontal force (thrust) that tries to push wall plates apart:
Tension members installed along the bottom plate level (ceiling joists). They physically tie opposing walls together to resist outward thrust. If ceiling joists are omitted for vaulted ceilings, an engineered structural load-bearing ridge beam must support the ridge in compression.
Installed in the upper third of the attic space (spaced max 4 ft O.C.). Collar ties do not prevent wall spreading; their sole structural purpose is resisting wind uplift suction and preventing the ridge from pulling apart during windstorms.
Hip and valley rafters run diagonally at $45^\circ$ across building corners. Because the diagonal of a 12×12-inch square is $\sqrt{12^2 + 12^2} = 16.97\text{"} \approx 17\text{"}$, hip rafters have a longer run and shallower pitch:
The horizontal run of a hip rafter is always $\sqrt{2} \approx 1.4142$ times the common rafter run ($\text{Hip Run} = \text{Common Run} \times 1.4142$).
A 6/12 pitch common rafter uses 6 on 17 on the steel framing square body to mark plumb cuts and level seat cuts on hip and valley rafters.
Because hip stock sits shallower, deepen the seat cut by 1/4" to 3/8" so the hip centerline matches common rafter Height Above Plate (HAP).
Jack rafters decrease in length at a constant mathematical rate ($\Delta L$) along the hip or valley:
To calculate common rafter length: (1) Find the Rafter Run: Run = (Total Building Span - Ridge Board Thickness) / 2. (2) Calculate Line Length: Line Length = Run × √(1 + (Pitch Rise / 12)²). For example, on a 24-foot building span with a 1.5-inch ridge board and a 6/12 pitch, Run = (288" - 1.5") / 2 = 143.25 inches. Line Length = 143.25 × √(1 + 0.25) = 143.25 × 1.1180 = 160.16 inches (13 feet 4-3/16 inches). (3) Add Overhang: Add the sloped overhang length to get the total lumber piece required.
Rafter Line Length is the theoretical working length measured from the long point of the plumb ridge cut to the plumb heel cut line of the birdsmouth notch. Total Board Length includes the Line Length plus the sloped eave overhang tail plus any extra material needed for fascia cuts. When ordering lumber from a home center, always size by Total Board Length rounded up to the nearest standard 2-foot commercial stock increment (8, 10, 12, 14, 16, 18, 20, 24 feet).
Hip and valley rafters run diagonally at 45 degrees relative to exterior walls. Because the diagonal of a 12x12-inch square is 16.97 inches (the 17-inch framing square rule), hip rafter run is Hip Run = Common Run × √2 = Common Run × 1.4142. The Hip Length is Hip Length = √(Rise² + Hip Run²). For example, with a 6/12 common pitch (6/17 on the hip), a 10-foot common run yields a 14.14-foot hip run, producing a 15.17-foot hip rafter line length (15 ft 2-1/16 in).
Jack rafters are common rafters that terminate against a hip or valley rafter rather than reaching the ridge. Jack rafters decrease in length by a uniform common difference (ΔL) based on their on-center spacing: ΔL = On-Center Spacing × (Common Rafter Length / Common Run) = Spacing × √(1 + (Pitch/12)²). For a 6/12 roof with 16-inch O.C. spacing, each successive jack rafter is exactly 16 × 1.1180 = 17.89 inches (17-7/8 in) shorter than the preceding one.
Under IRC Section R802.4.2, a ridge board must have a depth at least equal to the cut end of the rafters and typically has a nominal thickness of 1 inch (actual 0.75\") or 2 inches (actual 1.5\"). The ridge deduction is half the actual thickness of the ridge board (0.75\" for a 2x ridge board). This 0.75\" deduction must be measured horizontally or perpendicular to the plumb cut line at the top of each rafter to ensure opposite rafters meet flush without bowing the ridge.
Under IRC Table R802.5.1 (Douglas Fir-Larch #2, 20 psf live load, 10 psf dead load, 16\" O.C. spacing): (1) 2x4 Lumber: Max horizontal span is 9 ft 10 in; (2) 2x6 Lumber: Max horizontal span is 14 ft 4 in; (3) 2x8 Lumber: Max horizontal span is 18 ft 2 in; (4) 2x10 Lumber: Max horizontal span is 22 ft 0 in; (5) 2x12 Lumber: Max horizontal span is 25 ft 6 in. Note that rafter span is measured horizontally along the run, not along the sloped hypotenuse length.
