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Rafter Length Calculator

Lay out a common rafter from the run and pitch. Get its length with the ridge reduction and overhang, the cut angles, and the birdsmouth notch size.

Level: Grades 8–10

  • Common Core standard 8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
  • Common Core standard HSG.SRT.C.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Printable worksheet

Rafter Length Calculator

Know the answer you need? Pick an input to work backward.

Inputs

For a 6/12 roof enter 6.

Half the span for a symmetrical gable: a 24 ft wide building has a 12 ft run.

1.5 in for a 2× ridge board; 0 if rafters meet each other with no ridge board.

Usually the width of the top plate: 3.5 in for a 2×4 wall, 5.5 in for a 2×6 wall.

Actual size of the lumber's wide face: a 2×8 is 7.25 in, a 2×10 is 9.25 in.

Result

Rafter length (with overhang, ridge reduction taken)

173.57 in

Each common rafter is about 173.57 in (14 ft 5 9/16 in) long, cut plumb at 26.57°.

Rafter length in feet and inches
14 ft 5 9/16 inRounded to the nearest 1/16 in.
Line length (wall line to ridge face)
160.16 in
Tail length (overhang)
13.42 in
Ridge reduction along the rafter
0.839 in
Roof angle (plumb cut off square; seat cut is 90° minus this)
26.57 °
Seat (level) cut angle off square
63.43 °
Pitch factor
1.118
Rise from wall line to ridge center
72 in
Birdsmouth heel (plumb cut) height
1.75 in
Height above plate
6.356 in
Notch depth (square to the rafter edge)
1.565 in
Birdsmouth check (a rule of thumb; follow your code and engineer)
Notch is 21.6% of the rafter depth, within the common one-third guideline

See the step-by-step solution ↓

Step-by-step solution

  1. Find the pitch factor: how many inches of sloped rafter you need for each inch of level run

    1+(612)2=1.118034

  2. Turn the run into inches and take off half the ridge board, because rafters stop at its face, not its center

    12×12−1.52=143.25 in

  3. Multiply that run by the pitch factor to get the sloped length from the wall to the ridge

    143.25×1.118034=160.1584 in

  4. Multiply the overhang by the pitch factor too, since the tail slopes at the same angle

    12×1.118034=13.4164 in

  5. Add the main length and the tail to get the whole rafter

    160.1584+13.4164=173.5748 in

  6. Find the roof angle from the pitch. That angle sets both the top (plumb) cut and the seat cut

    arctan612=26.5651 °

  7. Find the height of the notch's upright cut (the heel): the seat length times the rise per inch

    3.5×612=1.75 in

  8. Find how much wood stays above the wall: the rafter's upright depth minus the heel

    7.25×1.118034−1.75=6.3557 in

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find how long to cut each rafter and the angles for every cut. A common rafter is a plain roof board that runs straight from the top of the wall (the wall plate) up to the ridge. It sits at a right angle to the wall. Framers lay one out as a pattern and cut the rest to match, so the pattern has to be right.

That pattern needs a few things. It needs the length from the wall line to the ridge, less half the ridge board, plus the tail that forms the overhang. It also needs a plumb (upright) cut at the top and a tail cut at the end. Where it sits on the wall, it needs a birdsmouth: a small notch shaped like an open beak.

This calculator works through that layout from the pitch (the roof's steepness) and the building run. It multiplies the run and the overhang by the pitch factor and takes off the ridge reduction. It gives the plumb and seat cut angles. It also sizes the birdsmouth, so you can check that the notch leaves enough wood.

Rafter size, spacing, span and connections are set by loads and by the building code adopted where you build. This is a layout aid. A licensed professional or the building official should approve the framing.

Where it's used

Why this math matters outside the classroom:

  • Pattern rafters. Framers cut one pattern rafter for a 6/12 roof with a 12 ft run, about 173.57 in long with a 12 in overhang, and copy it.
  • Shed roofs. A shed builder with a 6 ft run at 4/12 finds the line length as 72 × 1.0541 ≈ 75.9 in before the overhang.
  • Lumber ordering. Builders round a 14 ft 5 9/16 in rafter up to 16 ft stock, since the cuts need extra length.
  • Birdsmouth checks. Carpenters check that a 1.57 in notch on a 7.25 in deep 2×8 leaves well over two thirds of the depth.
  • Right triangle practice. Students see the rafter as the hypotenuse of a 12 ft run and 6 ft rise: √(12² + 6²) ≈ 13.42 ft.

Grade level and standards: Grades 8–10; Common Core State Standards for Mathematics 8.G.B.7, HSG.SRT.C.8. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Enter the roof pitch as inches of rise per 12 inches of run. Then enter the building run, from the outside of the wall to the center of the ridge.
  2. Enter the ridge board thickness and the level overhang past the wall.
  3. Enter the birdsmouth seat length, usually the width of the wall plate. Then enter the actual depth of the rafter lumber.
  4. Read the rafter length, the cut angles and the birdsmouth figures. Mark them on a pattern rafter.

What the inputs mean

Roof pitch: rise per 12 of run
For a 6/12 roof enter 6.
Building run (outside of wall to center of ridge)
Measure level, from the outside face of the wall framing to the center line of the ridge. For a gable roof (two slopes meeting at a center ridge), this is half the building's width.
Ridge board thickness
Rafters stop at the face of the ridge board, so half its thickness comes off the run. That happens before the pitch factor is applied. A 2× ridge is 1.5 in thick, which takes 0.75 in off the run.
Birdsmouth seat cut (bearing on the wall plate)
The seat is the level part of the birdsmouth that rests on the top plate. It usually equals the plate width. On steep roofs or shallow rafters, it's often shortened to keep the notch small.
Rafter depth (actual)
Use the lumber's real size, not its name: 2×6 = 5.5 in, 2×8 = 7.25 in, 2×10 = 9.25 in, 2×12 = 11.25 in.

Formula and method

pitch factor = √(1 + (pitch ÷ 12)²)
line length = (run × 12 − ridge ÷ 2) × pitch factor
tail length = overhang × pitch factor
rafter length = line length + tail length
roof angle = arctan(pitch ÷ 12)
heel height = seat cut × pitch ÷ 12
height above plate = rafter depth × pitch factor − heel height

In the formulas, pitch is the inches of rise per 12 inches of run, and run is in feet (times 12 turns it into inches). Ridge is the ridge board's thickness. Overhang, seat cut and rafter depth are in inches. The pitch factor is the sloped length for each inch of level run. It comes from the Pythagorean theorem, the rule for the long side of a right triangle.

Every level measurement gets multiplied by the pitch factor to give a length along the rafter. That means the run (less the ridge reduction) and the overhang.

The plumb cut at the ridge and the heel of the birdsmouth are upright lines. So they sit at the roof angle off a square cut across the rafter. The seat cut is level, at 90° minus that angle.

The birdsmouth forms a small right triangle. Its level side is the seat and its upright side is the heel. The height above plate is how much of the rafter's upright depth is left above the notch. Framers keep this figure the same on every rafter, so the roof surface stays flat.

Worked example

A 6/12 roof has a 12 ft run, a 2× ridge board (1.5 in), a 12 in horizontal overhang, and 2×8 rafters (7.25 in) sitting on a 2×4 wall (3.5 in seat).

  1. Pitch factor: √(1 + 0.5²) = 1.118034.
  2. Net run: 12 × 12 − 1.5 ÷ 2 = 143.25 in, so the line length is 143.25 × 1.118034 = 160.1584 in.
  3. Tail: 12 × 1.118034 = 13.4164 in, for a total of 173.5748 in, about 14 ft 5 9/16 in.
  4. Roof angle: arctan(6 ÷ 12) = 26.57°, so the plumb cut is 26.57° off square and the seat cut 63.43°.
  5. Birdsmouth: heel 3.5 × 6 ÷ 12 = 1.75 in; height above plate 7.25 × 1.118034 − 1.75 = 6.356 in.

Result: Each rafter is 173.57 in (about 14 ft 5 9/16 in) along its top edge, and the birdsmouth notch takes about 21.6% of the depth.

Understanding the result

The rafter length is measured along the top edge. It runs from the plumb cut at the ridge to the end of the tail, before any fascia (trim board) or tail-cut detail. Buy lumber at least a little longer than this. The plumb and tail cuts need extra wood beyond the measured line.

The birdsmouth check looks at the notch depth, measured square to the rafter's edge. A common rule of thumb says it should be no more than one third of the rafter depth. It's a guide only. The code adopted locally and the roof's designer set what is allowed.

Assumptions and limitations

  • It's a common rafter in a gable that's the same on both sides. Seen from above, it's square to the wall, with the run measured to the center of the ridge.
  • The overhang is measured level from the outside of the wall framing.
  • The lumber is the same depth all along, and the top plate is level.
  • Hip, valley and jack rafters run at an angle when seen from above, so they need different factors.
  • There's no structural check. It doesn't look at span, size, spacing, snow and wind loads, uplift ties (straps that hold the roof down in wind) or ridge support.
  • It doesn't lay out fascia, sub-fascia or tail-cut styles (plumb, square or level).

Frequently asked questions

What is the ridge reduction?

The rafters stop against the face of the ridge board, not at its center line. So half the ridge's thickness comes off the run. With a 1.5 in ridge, 0.75 in of run comes off. On a 6/12 roof, that shortens the rafter by about 0.84 in along its length.

What angle do I cut the top of a rafter?

The plumb cut sits at the roof angle off a square cut: arctan(pitch ÷ 12). Arctan is the button on a calculator that turns a slope into an angle. A 6/12 roof needs 26.57°, a 4/12 roof 18.43° and a 12/12 roof 45°. A rafter square or speed square marks the same line when you pivot it to the pitch number.

How deep can a birdsmouth be?

A common framing rule of thumb keeps the notch to no more than a third of the rafter depth. That leaves enough wood over the wall. The code that applies, engineered drawings and the inspector have the final word. Steep roofs often need a shorter seat cut to stay within it.

Is the rafter length the board length I should buy?

No. The calculated length runs along the top edge, from the ridge plumb cut to the end of the tail. The angled cuts need extra wood beyond that line. So round up to the next stock length, which usually comes in 2 ft steps.

How this calculator is checked

Method: Pythagorean pitch factor applied to the run less half the ridge and to the overhang; cut angles from arctan(pitch/12); birdsmouth as a right triangle with a level seat and plumb heel. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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