Hip / Valley Rafter Length Guide

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Why Hip / Valley Rafters Use Run × 1.4142

Look down on the building. Commons run perpendicular to the ridge. A hip runs to the outside corner; a valley runs to the inside corner. On a 90° corner that path is 45° in plan. The plan distance is the diagonal of a square whose side is the common run.

1.4142 is √2 rounded the way framers print it on a square (the 17-inch hip run: 12 × 1.4142 ≈ 16.97). Rise does not get the 1.4142. If you multiply the common hypotenuse by 1.4142 you pretend the roof got taller. It did not. That is the #1 way a hip comes out long.

The existing common vs hip/valley page is the decision article for this cluster. It already explained that triangle. This guide is how to trust the 1.4142 and then put it on a board. We did not write a second decision article.

How Plumb Cut and Seat Cut Come From Pitch

Common plumb is arctan(rise ÷ common run) — the roof pitch in degrees. Hip plumb is arctan(rise ÷ diagonal run). Same rise, longer run, flatter angle. Seat cut is 90° minus that hip plumb, the level cut that wants to sit on the plate.

On the 12 ft / 6/12 example, common plumb is 26.57°. Hip plumb is about 19.5°. If you set a Speed Square or a circular saw to 26.6° on a hip, the ridge joint is wrong even if the length was accidentally close. The calculator prints both so you can see the difference.

How to hold a square and tick a plumb vs a seat is already written for commons in the rafter length and pitch guide. Use that sequence. Swap in the hip numbers: 17 on the body (not 12), pitch on the tongue, hip plumb on the saw. Do not copy the common degree onto a hip.

How to Apply the Result on an Actual Hip / Valley

1. Get the common run right. Plate to ridge, horizontal, the same run the common calculator uses. A 24 ft centered gable is 12 ft of common run. Do not type the hip plan run into the hip calculator or you will apply 1.4142 twice.
2. Run the calculator. Open the Hip / Valley Rafter Length Calculator. Write diagonal run, centerline length, hip plumb, seat, and any thickness/overhang extras on the stock. PDF the cheat sheet if you want it on the sawhorse.
3. Crown up, pattern first. Sight the hip. It is a longer, often crowned 2x. Cut one pattern and prove it at the corner before you clone.
4. Square at 17 and pitch. Body on 16.97" (marked 17 on most squares), tongue on the roof pitch number. That walk is the hip plumb and the complementary seat. Stair gauges lock it.
5. Ridge plumb, then length. Tick hip plumb at the top. Measure centerline length down the measuring line. If you entered thickness, the calculator already shortened for a 45° ridge setback — do not subtract it again by eye.
6. Birdsmouth is skewed. The seat is still the level cut, but in plan the plate corner is 45° to the hip. The notch is not the same rectangle as a common. Remaining depth still matters; overcutting the crotch is the roof version of killing a stringer throat. Stairs use a different triangle: stair stringer layout guide — do not mix those walks on one square without resetting gauges.
7. Backing is a second layout. Jacks need a plane on the hip edge. That bevel is not the plumb number. Cross-check backing on the square or a rafter table. This calculator on purpose does not invent a backing angle.

What This Number Is Not

It is not a member-size stamp. IRC R802 still wants the right depth, species, and spans for the roof load. It is not jack lengths (jacks keep common plumb and step down in length). It is not permission to skip backing. It is not a substitute for the common calculator on sticks square to the ridge.

We skipped a hip/valley “pitch chart” page on purpose. A table of hip plumb vs common pitch would only restate what the calculator already prints live from your run. There is no independent reference grid here the way joist spans have IRC tables.

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