How do you calculate hip rafter length?

Find the hip run first: on an equal-pitch roof it's the common run × 1.414. Then find the rise from the pitch (run × pitch ÷ 12). The hip length is √(hip run² + rise²). For a 4/12 roof with a 24 in run that's 34.87 in, before deductions.

Why is the hip run 1.414 times the common run?

On an equal-pitch roof the hip runs at 45° in plan, along the diagonal of a square. The diagonal of a square is its side × √2, and √2 = 1.414.

How do you find the diagonal of a box?

Do the Pythagorean theorem twice. First the diagonal across the bottom: √(length² + width²). Then use that and the height: √(bottom diagonal² + height²). For a cube with side s the answer is s × 1.732.

Is hip rafter length the same as the common rafter length times 1.414?

No. The run is multiplied by 1.414, not the rafter. The rise stays the same, so the hip comes out shorter than common rafter × 1.414: on a 4/12 roof with a 24 in run, 25.30 × 1.414 is 35.78 in, but the real hip is 34.87 in.

MeasuringShop Math & AnglesHip rafter length

How do you find a hip rafter length without a construction calculator?

The construction calculator spits out a hip length and everybody nods like it's magic. Then the battery dies. Akribis leather's cube drawing shows what's actually inside the box: the Pythagorean theorem, done twice. Once you see it, you can work a hip on the back of an offcut. Or check the calculator when it gives you a number that feels wrong.

⚡ Rule of thumb

A hip is a 3D diagonal, so solve two right triangles in a row. First the plan: for an equal-pitch hip, hip run = common run × 1.414. Then stand a second triangle on that line: hip length = √(hip run² + rise²).

Starting point

Hip run = common run × 1.414 · hip length = √(hip run² + rise²) · 4/12, 24 in run → 34.87 in (886 mm)

Measured

Theoretical line length, before ridge and birdsmouth deductions; one unit throughout

Confidence

Widely accepted

▶ Source video — Can you calculate a hip without a construction master calculator? — Akribis leatherwatch on Facebook ↗

Applies to

Process: Roof framing layout: hips and valleys on equal-pitch roofs
Also works for: Any corner-to-corner diagonal through a box: braces, frame cross-members, cabinet carcasses
Setup: Phone calculator (√ key), pencil, paper
Inputs: Common run and pitch (rise per 12 in of run)

Why it matters

A hip doesn't run up the roof the way a common rafter does. It runs diagonally across the plan and up at the same time, so you can't get it from one triangle.

That's the part people miss.

Use the common rafter length as a leg and the hip comes out short every time. The fix is just order of operations: flat triangle first, then the standing one. And since it's the same geometry as any diagonal through a box, the method is worth more than the roof.

What good looks like

The plan diagonal (hip run) is worked out before the rise comes in
Everything stays in one unit, inches or millimetres, start to finish
Cube check: a cube with side s has a corner-to-corner diagonal of s × 1.732 (side 5 → 8.66)
Your hip length per foot of common run matches the rafter table on a framing square (17.44 for 4/12, 20.78 for 12/12)

How to apply it

Warm up on the flat triangle: A² + B² = C². The 3-4-5 check is 9 + 16 = 25, and √25 = 5.
Find the plan diagonal. For a square plan with side 5: √(5² + 5²) = √50 = 7.071.
Stand a second right triangle on that diagonal. The plan diagonal (7.071) is now its base, and the rise (5) is its height.
Solve again: √(7.071² + 5²) = √75 = 8.66. That's the corner-to-corner length through the cube.
On a roof, get the rise from the pitch first: rise = common run × pitch ÷ 12.
Get the hip run: common run × 1.414 (that's √2, the diagonal of a square).
Hip length = √(hip run² + rise²).
Worked example: 4/12 pitch, 24 in (610 mm) common run. Rise = 24 × 4 ÷ 12 = 8 in (203 mm). Common rafter = √(24² + 8²) = √640 = 25.30 in (643 mm). Hip run = 24 × 1.414 = 33.94 in (862 mm). Hip = √(33.94² + 8²) = 34.87 in (886 mm).

Test it on your setup

Known case: a 12/12 roof with a 12 in (305 mm) run has a common rafter of 16.97 in (431 mm) and a hip of 20.78 in (528 mm). That's 12 × √3, the cube diagonal. If your method gives that, it works.
Build a cardboard box, any size, and tape-measure it corner to corner through the inside. It should match your two-step math to within the thickness of the cardboard.
Check against the framing square: the "length of hip or valley per foot run" line on the blade gives the hip per 12 in of common run. 4/12 reads 17.44, and 17.44 × 2 ft = 34.87 in.

Exceptions & tradeoffs

The × 1.414 shortcut only holds when both roof planes share the same pitch, so the hip sits at 45° in plan. With unequal pitches, use the real plan diagonal: √(run₁² + run₂²), where each run is that side's run to the hip.

These are theoretical line lengths to the centreline. Real hips still need a deduction for half the ridge or common thickness, a side cut, and a birdsmouth, and those depend on your lumber.

Mind you, this is the layout math, not a framing course. If you've never cut a hip, the numbers are the easy part.

Common mistakes

Using the common rafter length as one leg of the hip triangleUse the hip run (the plan diagonal) and the rise. The common rafter is a different triangle
Mixing feet and inches between the two stepsConvert everything to inches (or millimetres) before step one
Rounding the plan diagonal hard before the second stepCarry at least two decimals. 34 in instead of 33.94 isn't much, but it compounds on a long hip

Frequently asked questions

How do you calculate hip rafter length?Find the hip run first: on an equal-pitch roof it's the common run × 1.414. Then find the rise from the pitch (run × pitch ÷ 12). The hip length is √(hip run² + rise²). For a 4/12 roof with a 24 in run that's 34.87 in, before deductions.
Why is the hip run 1.414 times the common run?On an equal-pitch roof the hip runs at 45° in plan, along the diagonal of a square. The diagonal of a square is its side × √2, and √2 = 1.414.
How do you find the diagonal of a box?Do the Pythagorean theorem twice. First the diagonal across the bottom: √(length² + width²). Then use that and the height: √(bottom diagonal² + height²). For a cube with side s the answer is s × 1.732.
Is hip rafter length the same as the common rafter length times 1.414?No. The run is multiplied by 1.414, not the rafter. The rise stays the same, so the hip comes out shorter than common rafter × 1.414: on a 4/12 roof with a 24 in run, 25.30 × 1.414 is 35.78 in, but the real hip is 34.87 in.

Akribis leatherSource & visual reference

The Pythagorean theorem applied twice through a cube (5 → 7.071 → 8.66), plus rise from pitch and a common rafter example in the caption — Can you calculate a hip without a construction master calculator with only a run length and pitch/slope? ↗ at 1:10.

Summarized and independently edited by Start Here from Akribis leather's clip and caption, read off the frames, no audio. The double-Pythagoras cube (5, 7.071, 8.66) and the 4/12 common-rafter example are his. He explains the idea; he doesn't work a real hip on screen. So the 1.414 hip run, the 34.87 in hip, the 12/12 check, the framing-square cross-check and the unequal-pitch and deduction cautions are mine. I re-ran every number.