Hip and valley rafter calculator

A hip or valley runs at 45° across the corner of the building, so for every 12 in a common rafter runs, it runs 16.97 in. Besides the length you get the side (cheek) cuts and how much to drop or back a hip so the sheathing lies flat. Sizes and supports come from your building code. New to this? Words explained.

Why a hip runs 16.97 for every 12

On an equal-pitch roof the hips and valleys run at 45° to the walls. While a common rafter moves 12 in toward the ridge, the hip moves 12 in in both directions at once — √(12² + 12²) = 16.97 in of level run — and climbs the same rise. So the hip uses the same pitch number as the commons, but over 16.97 instead of 12. That is the “length hip or valley per foot run” line of a framing square’s rafter table.

common run = building width ÷ 2 − ½ ridge thickness
hip length per foot of common run = √(16.97² + rise²)   (inches)
hip length = (common run + overhang) × √(16.97² + rise²) ÷ 12
hip slope = atan(rise ÷ 16.97)

Example: a 24 ft wide hip roof at 6/12 with a 1½ in ridge and a 12 in overhang. The common run is 144 − ¾ = 143¼ in. At 6/12 the hip runs √(16.97² + 6²) = 18.00 in for every foot of common run, a factor of 1.5, so the hip is 214.88 in from ridge to corner and the overhang adds 18 in along the hip: 232.88 in, about 19 ft 4-7/8 in. With 2×6 commons the hip must be a 2×8 or deeper, and a plumb-cut tail plus trim makes a 236.44 in board — 20 ft stock.

Laying out a hip rafter

1. Ridge cut. Mark the plumb line at the hip slope — on a speed square, pivot to the common pitch on the HIP-VAL scale; on a framing square, 17 on the blade and the rise on the tongue.
2. Shorten it. Move the plumb line back by half the 45° thickness of the ridge, measured level — about 1-1/16 in for a 1½ in ridge. (Framed between the end common rafters instead, use half the 45° thickness of a common.)
3. Cheek cuts. Cut the plumb line with the saw at a 45° bevel. A hip against the ridge board takes a single cheek cut; one between the end commons takes a double cut to a point.
4. Birdsmouth. Lay out the seat and heel with the same height above plate (HAP) as the commons, then deepen the seat by the drop — or keep the commons’ HAP and back the top edge.
5. Tail. Step the overhang off at the hip slope (it runs 1.414 times the commons’ overhang) and give the tail a double cheek cut so fascia meets it from both walls.

Backing or dropping a hip

A hip’s top edge is flat, but the two roof planes meet at its centre line. Left alone, its outside corners stick up above the jack rafters and the sheathing rocks. There are two fixes. Dropping lowers the whole hip: cut the birdsmouth seat deeper by half the hip’s thickness × rise ÷ 16.97 — about 3/8 in for a 1½ in hip at 8/12. Backing bevels both sides of the top edge to a shallow peak: mark a line on each face half the thickness × sin(hip slope) below the top edge and plane or rip down to it; the saw tilt for that bevel is atan(sin(hip slope)), 23.1° at 8/12. Dropping is the quicker fix; backing takes longer but leaves the sheathing bearing on the hip’s full width.

A valley is the opposite: its top corners already sit below the roof planes, so the Navy’s framing manual notes that a valley needs neither backing nor dropping. The sheathing bears on the jacks either side.

Valleys, ridges and supports

• Equal-span valleys run from the inside corner to the ridge, the same length as a hip on the same width, shortened by half the ridge’s 45° thickness.
• A narrower wing needs a long (supporting) valley to the main ridge and a short valley that frames against it; the short valley’s run is half the wing’s width.
• Hip roof ridge. On an equal-pitch hip roof the ridge line length is the building length minus its width. Framed with the hips against the ridge, the board runs longer at each end by half its thickness plus half the hip’s 45° thickness.
• Size and support. IRC R802.4.3: hips and valleys at least 2 in nominal thick and not less in depth than the cut end of the rafter, braced at the ridge to a bearing partition or designed to carry the load. Below 3/12, R802.4.4 makes ridges, hips and valleys beams. Long hips are often doubled or engineered (LVL) — a design question, not a geometry one.

The common rafters on the same roof come from the rafter length calculator, which also gives the HAP the hip must match.