How to Figure Rafter Length from Pitch and Run
Why this matters
A common rafter is cut from a pattern, and every rafter on that side of the roof copies the pattern. An error in the pattern is an error repeated twenty or forty times, and it shows up only when the pairs go up: plumb cuts that gap at the ridge, a ridge board that rides high and bows, birdsmouths that do not sit flat on the plate. Recutting a full set of rafters is a lost day and a lumber order.
The length comes from three numbers you already have or can measure: the span, the pitch, and the ridge thickness. This card turns them into a line length, adds the overhang, and checks the result two ways before the first board is cut. It applies to a simple gable with a common rafter on each side. Rafter size, spacing and the ridge detail are structural decisions that come from the span tables your code adopts or from the engineer, not from this arithmetic.
The vocabulary, in one section
HALF SECTION, COMMON RAFTER AT 6/12 (not to scale)
ridge center line
|
ridge board ->|=|
/ |=| rise to
/ |=| ridge center
measuring line / |=| = 14 ft x 6/12
/ |=| = 7 ft
/ |=|
tail / |=|
\ / |=|
\____________/_______________|=|
|<-16 in->|<----- run 14 ft ---->|
overhang ^ outside face of
(level) wall top plate
|<------ half span = 14 ft ----->|
(plate face to center)
- Span: the horizontal distance between the outside faces of the two wall top plates. A 28 ft span here.
- Run: half the span for an equal-pitch gable, measured level from the outside face of the plate to the ridge center line. 14 ft here.
- Rise: the vertical distance the rafter climbs over the run, at the ridge center. Pitch times run: 14 ft x 6/12 = 7 ft.
- Measuring line: the line along the rafter on which lengths are measured, running parallel to its edges and passing through the point where the seat cut of the birdsmouth meets the heel plumb cut, which sits on the outside face of the plate.
- Line length: the length of the measuring line from the plate face to the ridge plumb cut.
- Overhang: how far the tail projects past the wall, given as a level distance.
Step 1: Measure the span outside to outside, at both ends
Pull the tape across the top plates from outside face to outside face, at both gable ends and once in the middle. Walls are rarely perfectly parallel; if the readings differ, the rafters for each bay have to suit that bay, or the walls get pulled or pushed before the roof goes on.
Why outside to outside: the run is defined from the outside face of the plate, because that is where the measuring line meets the birdsmouth. What breaks if you skip it: measuring inside to inside on 2x4 walls loses the thickness of both walls, 3-1/2 in each, so the span reads 7 in short, the run 3-1/2 in short, and every rafter about 3.5 x 1.118 = 3.91 in short along the slope.
Step 2: Take half the span, then deduct half the ridge
The theoretical run, to the ridge center, is half the span: 28 ft / 2 = 14 ft = 168 in.
The rafter does not reach the ridge center. It stops at the ridge board's face, half the ridge's thickness short of the center. With a 1-1/2 in ridge board, the deduction is 3/4 in, measured level, so the actual run is 168 - 0.75 = 167.25 in.
Why level: the ridge deduction is a horizontal distance from the center line to the face, so it comes off the run, not off the sloped length. If you prefer to compute the full line length first and then deduct, the deduction has to be converted to slope the same way the run is: 0.75 in x 1.118 = 0.84 in along the rafter.
Step 3: Multiply the run by the pitch multiplier
The pitch multiplier converts a level distance into the matching distance along the slope. For a pitch of rise R in 12, the multiplier is the square root of 1 + (R / 12) squared. For 6/12: 1 + 0.25 = 1.25, and the square root of 1.25 is 1.118. See related: Roof Pitch Multipliers for Any Sloped Surface, which owns the table.
Two routes, same answer:
- Run first: 167.25 in x 1.118 = 186.99 in.
- Center first: 14 ft x 1.118 = 15.652 ft = 187.82 in, minus the 0.84 in slope deduction (0.8385 unrounded) = 186.99 in.
When the two routes agree to the hundredth, the deduction went in correctly. Convert to tape terms only now: 186.99 in is 15 ft 6.99 in, and 0.99 x 16 = 15.8 sixteenths, which rounds to 16/16, so 15 ft 7 in on the measuring line. See related: How to Convert Decimal Feet to Feet, Inches and Sixteenths.
Step 4: Add the overhang the same way
The overhang is given level, so it gets the same multiplier. A 16 in overhang: 16 in x 1.118 = 17.89 in along the measuring line, from the plate face out to the tail plumb cut.
Confirm what the 16 in is measured to before you multiply. "16 in overhang" on one set of plans means wall face to the back of the fascia, where the tail cut lands, and on another it means wall face to the front of the finished fascia. In the second case, subtract the fascia thickness, which is also a level distance, from the 16 in first, then multiply.
Step 5: Total it and pick the stock length
Measuring line total: 186.99 + 17.89 = 204.88 in, which is 17.07 ft.
The board is longer than the measuring line, because the plumb cut at the ridge and the tail cut both extend past the line toward the board's top edge, and a saw cut needs a little extra to square the ends. A 16 ft board (192 in) is short by over a foot. An 18 ft board (216 in) leaves about 11 in for cuts and the long points. Order 18 ft stock.
Step 6: Lay out a pattern rafter and test it
Mark the pattern with a framing square set to 6 on the tongue and 12 on the blade, the same 6/12 the roof is built to.
- Mark the ridge plumb cut near one end, along the tongue of the square.
- Measure 15 ft 7 in down the measuring line from that cut and mark the heel plumb line of the birdsmouth.
- Mark the seat cut level from the heel line, along the blade. How deep the seat may be cut into the rafter is governed by your code's notching rules or the engineer, not by this layout.
- Measure 17-7/8 in further along the measuring line for the tail plumb cut (17.89 in, 0.89 x 16 = 14.2 sixteenths, so 17-14/16, which is 17-7/8).
- Cut the pattern and a second rafter from it, and test the pair at both ends of the building on the plates against a scrap of ridge before cutting the rest.
Why test a pair: it checks the span, the deduction and the plumb angle at once. The plumb cuts should close tight on the ridge, the seats should bear flat on the plates, and the tails should line up with a string pulled along the wall. Any gap tells you which number to recheck before the error is multiplied across the stack.
Cutting hazards apply at this step: eye and hearing protection on the saw, the board supported on both sides of the cut so the offcut cannot pinch the blade, and a dust control for the sawdust. Setting the pair on the plates is work at height: fall protection under your employer's program, which for construction work at 6 ft or more is governed by 29 CFR 1926.501.
Check the answer before the cut
Rebuild the triangle. Rise to ridge center is 14 ft x 6/12 = 7 ft. The center line length should be the square root of 14 squared plus 7 squared: 196 + 49 = 245, and the square root of 245 is 15.65 ft. That matches step 3.
The rough bracket. A rafter on any normal pitch is longer than its run and shorter than 1.414 times it, which is the 12/12 multiplier. Here that brackets the line length between 14 ft and 19.8 ft. 15.65 ft sits inside, and close to the low end, as a moderate 6/12 should.
The pitch check. The rise over the line should come back to the pitch: 7 ft of rise over 15.65 ft of slope has a sine of 0.447, an angle of 26.57 degrees, and tan(26.57) is 0.50, which is 6/12.
The errors and what each one does
Ridge deduction forgotten. Every rafter is 0.84 in long on the slope. With the seats held on the plates, the pair can only fit by lifting the ridge: on these numbers, a rafter 187.82 in long over the 167.25 in actual run climbs to the ridge face about 85.46 in, against 83.62 in for the correct rafter, so the ridge rides about 1-13/16 in high, the pitch steepens slightly, and the ridge plumb cuts tip off plumb, touching at the bottom and opening at the top.
The whole ridge thickness deducted instead of half. Each rafter comes out a further 0.84 in short on the slope. The pair now has to drop the ridge or leave a gap at the ridge face.
The overhang multiplied by the wrong pitch, or not at all. A 16 in overhang laid out as 16 in on the slope puts the tail 1.89 in short along the slope, about 1.69 in short of the fascia line measured level, and on a long wall the fascia is either shimmed out or the tails are recut one by one.
Rounding the multiplier. Using 1.12 for 1.118 on this rafter adds 167.25 x 0.002 = 0.33 in, which is small on one rafter and fine to ignore if every rafter is cut from the same pattern. Carry three decimals anyway and the question never comes up.
What changes the method
- Unequal pitches. Each side has its own run, and the two runs are not half the span. Find where the ridge sits from the two pitches and the total span, then figure each side from its own run.
- A structural ridge beam. Rafters hung on hangers on a beam's face take the same half-thickness deduction, using the beam's width. Rafters that bear on top of a beam are laid out with a seat cut at the beam instead, which is a different layout.
- Hips and valleys. They run diagonally in plan, so they use a different unit run and a different multiplier, and their length is a separate calculation. Do not apply the common rafter multiplier to them.
References
- See related: Roof Pitch Multipliers for Any Sloped Surface
- See related: Roof Slope, Pitch, + Measurement Reference
- See related: Write the Calculation Down: The Field Math Habit
- 29 CFR 1926.501, duty to have fall protection, for construction work