How do you calculate the length of a hip?

Multiply the common run by √(2 + (rise ÷ 12)²). At 8/12 that factor is 1.563, so a 16 ft run gives a 25.0 ft hip.

Is a valley calculated the same way as a hip?

Yes, when both planes share the pitch and meet at a square corner. Use the same factor with the common run of the wing the valley closes.

Can I use the pitch factor for a hip?

No. The pitch factor gives a common rafter’s length. A hip runs diagonally in plan, so it needs the hip / valley factor, which is always larger.

What if the two roof planes have different pitches?

Then this formula doesn’t apply, because the hip no longer sits at 45° in plan. Measure that hip on the roof or trace it from imagery.

Does the hip length include the overhang?

It does if you measure the run to the drip edge. Measure it to the wall plate and you get the length over the framing only.

Free calculator

Hip and Valley Calculator with true lengths from pitch and run

Enter the pitch, the level run each hip and valley covers, and the ridge length. You get every hip and valley length, the cap lineal feet and bundles, and the valley metal pieces.

The formula

The two planes on either side of the hip or valley share the same pitch, and the corner is square in plan. Factors are rounded to 3 decimals.

Hip / valley factor: √(2 + (rise ÷ 12)²) The 2 comes from the diagonal: the hip covers the run in both directions at once.
Hip or valley length: common run × hip / valley factor The true length along the roof, from the eave corner to the top of the line.
Cap lineal feet: ridge + all hips Valleys take no cap.
Cap bundles: ⌈cap lf ÷ 25⌉ 25 lf per bundle is typical; 20–35 lf is common, so check the wrapper.
Valley metal: ⌈valley length ÷ 10⌉ × number of valleys Whole 10 ft pieces, counted valley by valley.

Worked example

An 8/12 hip roof on a house 50 ft long and 32 ft wide, with a gable-ended wing 24 ft wide joining one side. The main roof has 4 hips, the wing makes 2 valleys and has 14 ft of ridge.

Hip / valley factor at 8/12: √(2 + (8 ÷ 12)²) = √2.444 = 1.563
Hip common run: 32 ÷ 2 = 16 ft
Length per hip: 16 × 1.563 = 25.0 ft
Total hips: 4 × 25.0 = 100.0 ft
Valley common run: 24 ÷ 2 = 12 ft
Length per valley: 12 × 1.563 = 18.8 ft
Total valleys: 2 × 18.8 = 37.6 ft
Ridge: main roof 50 − 32 = 18 ft, plus 14 ft on the wing = 32 ft
Cap lineal feet: 32 ft of ridge + 100.0 ft of hips = 132.0 lf
Cap bundles: 132.0 ÷ 25 = 5.28, rounded up to 6 bundles
Valley metal: ⌈18.8 ÷ 10⌉ = 2 pieces per valley × 2 valleys = 4 pieces

Four 25.0 ft hips and two 18.8 ft valleys: order 6 bundles of cap for 132 lf and 4 pieces of 10 ft valley metal.

Hip and valley factor by pitch

Equal pitches on both planes, square corner. The last column is the true hip or valley length for every 10 ft of common run.
Pitch	Pitch factor	Hip / valley factor	Per 10 ft of run
3/12	1.031	1.436	14.4 ft
4/12	1.054	1.453	14.5 ft
5/12	1.083	1.474	14.7 ft
6/12	1.118	1.500	15.0 ft
7/12	1.158	1.530	15.3 ft
8/12	1.202	1.563	15.6 ft
9/12	1.250	1.601	16.0 ft
10/12	1.302	1.641	16.4 ft
11/12	1.357	1.685	16.9 ft
12/12	1.414	1.732	17.3 ft

Why a hip is longer than a rafter

A common rafter climbs straight up the slope, so its length is the run times the pitch factor. A hip doesn’t. It runs at 45° in plan, across the corner where two roof planes meet, so it covers the run in both directions before it reaches the ridge.

Take a 16 ft run at 8/12. Seen from above, the hip crosses 16 × √2 = 22.6 ft. It climbs the same 10.7 ft as the rafters (16 × 8 ÷ 12). Put the two together and you get about 25.0 ft, which is 16 × 1.563. The common rafter over that run is only 16 × 1.202 = 19.2 ft. Use the pitch factor on a hip and you come up 5.8 ft short on every one.

Finding the common run

The common run is the level distance the hip or valley covers in one direction. On a plain hip roof it’s half the building’s width: 16 ft on a 32 ft wide house. Measure the width at the drip edge if you want lengths for cap and metal; measure at the wall plate if you’re framing.

For a valley, use half the width of the wing whose ridge runs into the main roof. A 24 ft wide wing gives a 12 ft run. On a straight hip roof, the ridge is the length minus the width: 50 − 32 = 18 ft.

Cap and valley metal

Cap covers the ridge and every hip, never the valleys. Add them up and divide by the coverage on the bundle. 25 lf per bundle is a typical figure, but 20–35 lf is common, so check the wrapper and redo the division if yours differs.

Valley metal is counted in 10 ft pieces, rounded up for each valley on its own. An 18.8 ft valley takes 2 pieces. If your metal laps at the joints, add the laps to the valley length before you divide. Closed-cut and woven valleys don’t use metal, so set the count to 0 for those.

Where the formula stops

The formula assumes both planes share the pitch and meet at a square corner. Unequal pitches, angled wings and bay roofs break that, and so does a hip that ends against a dormer or a lower roof. Those lines have to be measured.

For framing, a hip rafter’s cut length also depends on the plate run and the deductions at the ridge. The calculator gives the line length on the roof, which is what cap and metal are ordered from.

If the roof is cut up, trace it instead: RoofStruct gives you free, unlimited self-serve measuring and PDF reports, no card, with every hip, valley and ridge measured along the slope.

Hip and valley questions

How do you calculate the length of a hip?: Multiply the common run by √(2 + (rise ÷ 12)²). At 8/12 that factor is 1.563, so a 16 ft run gives a 25.0 ft hip.
Is a valley calculated the same way as a hip?: Yes, when both planes share the pitch and meet at a square corner. Use the same factor with the common run of the wing the valley closes.
Can I use the pitch factor for a hip?: No. The pitch factor gives a common rafter’s length. A hip runs diagonally in plan, so it needs the hip / valley factor, which is always larger.
What if the two roof planes have different pitches?: Then this formula doesn’t apply, because the hip no longer sits at 45° in plan. Measure that hip on the roof or trace it from imagery.
Does the hip length include the overhang?: It does if you measure the run to the drip edge. Measure it to the wall plate and you get the length over the framing only.