How to Find Rafter Length From Roof Pitch and Span

Geometry Lessons

The short answer

rafter = √(run² + rise²)

Halve the span to get the run, multiply the run by the pitch over twelve to get the rise, then take the square root of run squared plus rise squared. A 24 foot span at 6 in 12 pitch gives a 12 foot run, a 6 foot rise, and a rafter of √(144 + 36) = 13.42 feet, measured along the slope before any overhang is added.

The rafter is the hypotenuse of the roof. Everything about its length is Pythagoras on a triangle whose legs are the run, half the building span, and the rise, which the pitch dictates. Framers have carried this triangle on the tongue and body of a framing square for two centuries; the arithmetic below is the same triangle without the tool. Get the run right, translate the pitch honestly, and the rafter follows in one line.

Pitch is a slope, not an angle

Roof pitch in North American practice is quoted as inches of rise per twelve inches of run. A 6 in 12 roof climbs six inches for every horizontal foot. That makes the rise easy: rise = run × pitch / 12. For a 12 foot run at 6 in 12, the rise is 12 × 6/12 = 6 feet. No trigonometry has happened yet, and none is needed for the length.

The angle, if you want it, is the arctangent of pitch over twelve, about 26.6° for 6 in 12. It matters for saw settings on plumb and seat cuts, but the rafter length itself never asks for it. Length comes from the legs, and the legs come from span and pitch directly.

1

Run

Half the span: 24 / 2 = 12 feet from wall plate to the center line.

2

Rise

Run times pitch over twelve: 12 × 6/12 = 6 feet of climb.

3

Pythagoras

√(12² + 6²) = √180 = 13.42 feet along the slope, line length before overhang.

The roof triangle, dimensioned

The diagram sections the roof and highlights one rafter as the hypotenuse over its 12 foot run and 6 foot rise, with the pitch badge showing the 6 in 12 slope and the card walking Pythagoras through 144 plus 36 to 13.42 feet.

A roof section with a 24 foot span, a 12 foot run and a 6 foot rise at 6 in 12 pitch, the rafter highlighted as the hypotenuse and Pythagoras working through 144 plus 36 to reach 13.42 feet.
Run and rise are the whole input. The 13.42 feet is the line length to the ridge center; ridge board thickness and overhang adjust it on the saw horses, not in the formula.

Run is half the span

Using the full span as the run computes a rafter for a roof twice as wide. It is the most common blunder in quick mental takeoffs.

√(24² + 6²)givesfull span used24.74 ft

That rafter would overshoot the ridge by more than eleven feet.

√(12² + 6²)giveshalf span used13.42 ft

Each rafter covers only its own half of the building.

A worked check you can repeat on paper

A 24 foot wide garage gets a 6 in 12 gable roof. Run: 12 feet. Rise: 6 feet. Rafter: √(12² + 6²) = √180 = 13.416 feet, call it 13 feet 5 inches of line length.

Check it a second way with the unit rafter table stamped on any framing square. At 6 in 12, the length per foot of run is √(12² + 6²)/12 = 13.416/12 = 1.118 feet of rafter per foot of run. Twelve feet of run times 1.118 gives 13.42 feet, the same answer by the carpenter route. Two methods, one number, and the second one is how the answer gets checked at the saw without a calculator.

Practical adjustments, applied after the line length: subtract half the ridge board thickness measured along the slope, and add the overhang, also measured along the slope. A 16 inch horizontal overhang at 6 in 12 adds 16 × 1.118 = 17.9 inches of tail, not 16.

Steeper costs more than it looks

The rafter grows slowly at shallow pitches and quickly at steep ones. At 4 in 12 the multiplier per foot of run is 1.054, at 6 in 12 it is 1.118, at 9 in 12 it is 1.25, and at 12 in 12 it is 1.414. Going from 6 in 12 to 12 in 12 on the same span adds 26 percent to every rafter, and the same percentage to the sheathing and shingles above them, because roof surface area follows the slope length.

That multiplier is worth keeping on a card for estimating: roof area equals the flat plan area times the pitch multiplier. A 30 by 40 plan under a 6 in 12 roof carries 1200 × 1.118 = 1342 square feet of actual roof surface per the whole footprint, before overhangs.

Mistakes that catch people out

  • Using the span as the run. The run is half the span for a symmetric gable. The full span belongs to the building, not to either rafter.
  • Adding a horizontal overhang without converting. Overhang is specified horizontally but cut along the slope. Multiply the horizontal overhang by the pitch multiplier before adding it to the rafter.
  • Ignoring the ridge board. The line length runs to the center line of the ridge. Deduct half the ridge thickness, measured square to the plumb cut, or every rafter comes up proud.
  • Treating pitch as degrees. A 6 in 12 roof is 26.6°, not 6°. Feeding pitch numbers into trig functions as if they were angles produces quiet nonsense.
  • Rounding the multiplier to 1.1. Over a 12 foot run that rounding shorts the rafter by more than two inches. Carry 1.118, round the cut, not the constant.

Try these

  1. Span 28 ft, pitch 6 in 12. Find the rafter line length.
  2. Span 24 ft, pitch 4 in 12. Find the rafter.
  3. Span 24 ft, pitch 12 in 12. Find the rafter.
  4. Run 12 ft, 16 in horizontal overhang at 6 in 12. Find the slope length of the tail.
  5. A 30 by 40 ft plan at 9 in 12. Find the roof surface over the footprint.
Show the answers
  1. run 14, rise 7, √245 = 15.65 ft
  2. run 12, rise 4, √160 = 12.65 ft
  3. run 12, rise 12, √288 = 16.97 ft
  4. 16 × 1.118 = 17.9 in
  5. 1200 × 1.25 = 1500 sq ft

Common questions

How do you calculate rafter length from span and pitch?

Halve the span to get the run, multiply the run by the pitch over twelve to get the rise, then take the square root of run squared plus rise squared. That is the line length along the slope to the ridge center line.

What does a 6 in 12 pitch mean?

Six inches of vertical rise for every twelve inches of horizontal run. As an angle it is about 26.6 degrees, though the length calculation never needs the angle.

Is the run the same as the span?

No. For a symmetric gable roof the run is half the span, because each rafter only travels from its wall plate to the ridge in the middle.

How is the overhang added?

Along the slope. Multiply the horizontal overhang by the pitch multiplier, the rafter length per foot of run, and add that to the line length. Sixteen horizontal inches at 6 in 12 becomes about 17.9 inches of tail.

What about the ridge board?

The calculated length runs to the ridge center line. Deduct half the ridge board thickness so the rafter seats against the board rather than crowding it.

How much longer are rafters on a steeper roof?

By the pitch multiplier: 1.054 at 4 in 12, 1.118 at 6 in 12, 1.25 at 9 in 12 and 1.414 at 12 in 12. The same multiplier scales the roof surface area over a given footprint.

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