Hip Roof Pitch and Area: Hip Rafters, Jacks and the 17-Inch Rule

Hip roof geometry versus a gable

A gable roof has two rectangular planes and two vertical triangular end walls. A hip roof replaces those end walls with two more sloped planes, so all four sides of the building have a roof plane and none of them are vertical. Where a gable meets a gable end in a straight vertical line, a hip roof meets itself along a diagonal line called the hip — running from each corner of the building up to the ridge (or to a single point, on a square hip roof with no ridge at all, called a pyramid hip).

Because the hip line runs diagonally across the building's plan, it is not simply following one pitch. It is the intersection of two roof planes that meet at the corner, each sloping at the building's common pitch but running in two different directions in plan. That intersection line is both longer than a common rafter of the same run and set at a shallower slope angle than the roof's stated pitch, even though the roof surfaces themselves are at the common pitch. Framers handle this with two well-established shortcuts: the 17-inch rule for run, and the hip/valley factor table for length.

The 17-inch rule explained

On a standard hip roof with equal pitches on all sides and 45-degree corners, the hip rafter's run for every 12 inches of common rafter run is 12*sqrt(2), which equals 16.9706 inches — rounded up to 17 in nearly every framing square table and pattern rafter book. This comes directly from the Pythagorean theorem: the hip runs along the diagonal of a square whose two sides are each equal to the common run, so hip run = sqrt(run^2 + run^2) = run*sqrt(2).

This rule does two things at once. First, it gives you the hip rafter's own 'unit run' for laying out the seat and plumb cuts on a framing square or speed square: instead of stepping off cuts using 12 inches of run per foot like a common rafter, you step off using 17 inches of run per foot of rise, keeping the same rise number. Second, it tells you that the hip rafter's rise-to-run ratio, and therefore its slope angle, is shallower than the common rafter's stated pitch, even on the exact same roof.

Worked example: a 6:12 common pitch. The hip rafter's unit rise is still 6 inches, but its unit run is 17 inches instead of 12. The hip's own slope is arctan(6/17) = 19.44 degrees, versus the common rafter's arctan(6/12) = 26.57 degrees. The hip is visibly flatter along its own length even though the roof planes on either side of it are both cut at a true 6:12.

The hip and valley length factor

To get an actual hip or valley rafter length rather than just its unit run, use the length factor: factor = sqrt(rise^2 + 2*run^2) / run, where rise and run are the common rafter's rise and run for one foot of horizontal common run (run = 12). Multiply that factor by the total common run of the building (half the building width, in feet) to get the hip rafter's theoretical length, measured to the outside wall line before overhang and ridge/birdsmouth adjustments.

This is the same relationship as the 17-inch rule, expressed as a per-foot length rather than a per-foot run. It works because the hip rafter is the hypotenuse of a right triangle whose legs are the vertical rise and the diagonal horizontal run of 12*sqrt(2), so length = sqrt(rise^2 + (12*sqrt(2))^2) = sqrt(rise^2 + 288). Dividing by 12 (run in feet) gives the standard per-foot factor.

Jack rafters and the common difference

Jack rafters fill the triangular gaps between the hip rafter and the wall plate. They run parallel to the common rafters and share the same pitch and unit rise, but each one is shorter than the last as it gets closer to the corner, because the hip rafter cuts diagonally across the plan.

The amount each jack shortens from its neighbor is called the common difference, and it depends only on rafter spacing and the common rafter's length factor: common difference = (spacing in inches / 12) x common rafter length factor x 12, or more simply, common difference in inches per foot of spacing equals the common rafter factor times the spacing. For 16-inch on-centre spacing at a 6:12 pitch (factor 1.1180), the common difference is (16/12) x 1.1180 x 12 = 17.89 inches, so each successive jack is about 17 and 7/8 inches shorter than the one before it.

• The longest jack (nearest the wall) is cut first; each subsequent jack is shortened by the common difference and the pattern repeats identically on all four corners of a square-cornered hip roof.
• Jacks are cut with a single plumb cut at the top where they meet the hip, angled in plan at 45 degrees (a compound cheek cut) so the jack's end face sits flush against the hip rafter's side.
• Because the cheek cut angle depends on pitch, steeper roofs need a steeper cheek bevel; framing squares and rafter tables give this as the 'side cut' setting, read directly off the same tables used for the hip length factor.
• On hip roofs with a ridge (not a pyramid), jacks run in pairs from both the hip and, at the ridge end of a hip-and-valley intersection, may also be cut against a valley using the same common-difference logic.

Cheek cuts, bevels and the double side cut

A hip rafter itself needs a compound cut where it lands on the wall plate (the seat cut, which is a birdsmouth angled in both plan and elevation), and at the top where it meets the ridge board or the opposite hip (the cheek cut, a double bevel because two adjacent roof planes both terminate against the hip's two side faces).

The cheek cut on a hip rafter is commonly laid out using the same 17-unit method: instead of squaring the plumb line across the full width of the rafter stock, the line is dropped back half the stock thickness on each face, splitting the ridge intersection so both adjacent roof planes get full bearing. Rafter square tables list this as the 'hip/valley side cut' figure, typically expressed as inches per foot of run on the square's body and tongue, and it changes with pitch just as the hip length factor does.

Calculating hip roof area

Despite the more complex rafters, hip roof area uses the identical formula to a gable: sloped area = flat footprint area x pitch multiplier, where the pitch multiplier is sqrt(rise^2 + 12^2)/12 for the common pitch. The hip geometry changes how the material is cut and assembled on the roof plane, not the total area of sloped surface, because the four hip triangles and the two (or more) trapezoidal main planes still add up to exactly the same footprint projected at the same slope.

Worked example: a 40 x 30 foot hip roof at 6:12 with a 1-foot overhang on all sides. Footprint including overhang = 42 x 32 = 1,344 sq ft. Pitch multiplier at 6:12 = sqrt(6^2+12^2)/12 = sqrt(180)/12 = 1.1180. Roof area = 1,344 x 1.1180 = 1,502.6 sq ft = 15.03 squares.

The only extra step for a hip roof is confirming the plan is a true hip (45-degree equal-pitch corners) before applying the simple footprint method; irregular hip roofs with unequal pitches or non-45-degree corners need each plane measured and totalled separately, because the single-multiplier shortcut assumes symmetry that an irregular hip does not have.

Waste factors: hip roofs versus gables

A gable roof typically carries 10 percent waste for asphalt shingles, covering starter courses, ridge cap and normal offcuts. A hip roof needs more, because every hip line consumes cut shingles or panels at an angle, four corners need hip cap instead of two gable rakes, and jack courses generate more offcuts than the long, uninterrupted courses of a gable field.

For the 40x30 hip example above at 15.03 squares, apply 15 percent waste: 15.03 x 1.15 = 17.3 squares, rounded up to 18 squares of shingles (54 bundles at 3 bundles per square). The same footprint as a simple gable at 10 percent waste would only need 16.5 squares, so the hip geometry alone accounts for roughly a 1.5 to 2 square difference in material on a job this size — a real cost difference that estimators need to price in up front rather than discover mid-job.

1. Measure or calculate the flat footprint including all overhangs.
2. Apply the common-pitch multiplier to get true sloped area.
3. Add hip-specific waste (12-20% depending on pitch and complexity) instead of the standard gable 10%.
4. Order hip and ridge cap separately — it is sold by the linear foot, not by the square, and a four-hip roof needs roughly 4 x (hip length) plus ridge length of cap material.
5. Round up to the next full bundle count, then round up again to a full square if the job is tight on delivery logistics.

Frequently asked questions