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Woodworking Math

Roof pitch calculator

Give the pitch any way you have it: rise in 12, a rise and run you measured, an angle or a slope in percent. You get every other form at once, and below it the common rafter for your building: its length with the ridge and overhang, the plumb and seat cuts, a birdsmouth within the code's notch limit, and the span the International Residential Code allows for your lumber and snow load.

The pitch
The rafters

Over 70 psf of ground snow is past the IRC tables and needs a design.

Dead load
Ceiling
More: ridge, wall plate, raised ties, rafter count
Blank: at the wall top. Higher ties shorten the allowed span (IRC Table R802.4.1(9)).

Roof pitch

6/12

Angle
26.57°
Slope (grade)
50 %
Rafter per foot of run
13.416 in
Pitch as rise ÷ span
1/4

Geometry, exact: angle = atan(6 ÷ 12), slope = 6 ÷ 12 × 100, rafter per foot = √(12² + 6²).

Common rafter length

14 ft 5 9/16 in

On the measuring line: 13 ft 5 in line length, less 13/16 in for the ridge, plus 13 7/16 in of tail. Cut from a 16 ft 2x6.

Rafter
2x6 at 16 in
Allowed span
14 ft 7 in
Plumb cut
63.4° to the edge
Seat cut
26.6° to the edge
Birdsmouth seat
3 1/16 in
Heel (plumb) cut
1 9/16 in
Ridge board, at least
1x8 or 2x8
Rafters
Add a ridge length

One source: IRC 2024 and 2021, the model building code

Table R802.4.1(1) (Roof live load = 20 psf, ceiling not attached to rafters, L/Δ = 180): a Douglas Fir-Larch No. 2 2x6 at 16 in, 10 psf dead load, spans 14 ft 7 in. This rafter spans 11 ft 11 1/4 in level.

Framing square: 12 on the body, 6 on the tongue; mark the plumb cut along the tongue and the seat along the body. Speed square: pivot to 6 on the common scale.

  • OK: Seat notch 1 3/8 in deep, within a quarter of the 2x6’s depth (1 3/8 in; IRC R802.7.1 and R502.8.1): the seat is held to 3 1/16 in, shorter than the 3 1/2 in plate, and still bears more than 1 1/2 in (R802.6). Height above the plate 4 5/8 in.
  • OK: Overhang 12 in: within R802.7.1.1’s 24 in for a notched tail, with 4 1/8 in of rafter left over the plate.

The IRC tables cover sawn 2x4 to 2x12 rafters in four species groups, a 20 psf roof live load or up to 70 psf of ground snow, and a 10 or 20 psf dead load. Heavier loads, other species, wind uplift and trusses need a design. Check your local building code: your building department sets the snow load and the edition in force.

Why: A 2x6 rafter, 14 ft 5 9/16 in long on the measuring line, within Table R802.4.1(1).

Half the roof, seen from the gable end

The rafter in the accent colour from its tail to the ridge; dashed, the run (12 ft) across to the ridge’s centre and the rise (6 ft) up to it; the wall at the left.

IRC rafter spans for this lumber

Table R802.4.1(1), Douglas Fir-Larch No. 2, 10 psf dead load: the most each rafter may span, measured level
Spacing2x42x62x82x102x12
12 in10 ft 10 in16 ft 10 in21 ft 4 in26 ftover 26 ft
16 in9 ft 10 in14 ft 7 in (chosen)18 ft 5 in22 ft 6 in26 ft
19.2 in9 ft 1 in13 ft 3 in16 ft 10 in20 ft 7 in23 ft 10 in
24 in8 ft 2 in11 ft 11 in15 ft 1 in18 ft 5 in21 ft 4 in

“Over 26 ft” is the code’s Note b. A misprint is a cell the printed table gets wrong (it breaks the table’s own pattern), so it is not used.

Why these numbers?

  1. The situation

    A 6/12 roof over 24 ft, Douglas Fir-Larch No. 2 rafters 16 in on centre, 10 psf dead load.

    From your input: the roof and its lumber

  2. What that means

    Every foot of run takes √(12² + 6²) = 13.42 in of rafter, so each inch of run is 1.118 in along the slope.

    Arithmetic: exact by definition

  3. So

    Run 12 ft × 1.118 = 161 in, less 0.84 in for half the ridge, plus 13.42 in of tail = 173.57 in (14 ft 5 9/16 in).

    Arithmetic: exact by definition

  4. What that means

    IRC Table R802.4.1(1) (Roof live load = 20 psf, ceiling not attached to rafters, L/Δ = 180): a 2x6 spans 14 ft 7 in, measured level (R802.4.1).

    One source: International Code Council, the model building code , sources [1, 2]

  5. So

    The rafter spans 11 ft 11 1/4 in level, from the outside of the wall to the ridge’s face, within 14 ft 7 in.

    Arithmetic: exact by definition

  6. The number

    A notch at the end of a rafter may take up to a quarter of its depth (R802.7.1 and R502.8.1): 1.375 in for a 2x6. The rafter keeps at least 1 1/2 in of bearing (R802.6), and the ridge board is at least 1 in nominal and as deep as the 6.15 in plumb cut (R802.3).

    One source: International Code Council, the model building code , sources [1, 2]

  7. The number

    The IRC tables cover sawn 2x4 to 2x12 rafters in four species groups, a 20 psf roof live load or up to 70 psf of ground snow, and a 10 or 20 psf dead load. Heavier loads, other species, wind uplift and trusses need a design. Check your local building code: your building department sets the snow load and the edition in force.

    From your input: where you build

How we source these numbers

Pitch, angle and slope are one number written three ways

A roof's pitch is how far it rises for each 12 inches it runs level: a 6/12 roof climbs 6 inches every foot. The angle is the arctangent of rise over run, so 6/12 is atan(6 ÷ 12) = 26.57°. The slope or grade is the same ratio as a percentage, 6 ÷ 12 = 50 %. To measure one, hold a level 12 inches long against the rafter or the roof and measure straight up from its end to the roof: that height is the rise in 12. Any rise and run you measure give the same answer, since the calculator scales them to 12.

The older carpenters' “pitch” is the rise over the whole span, so a 6/12 roof is a quarter pitch (6 inches of rise on 24 inches of span) and a 12/12 roof is a half pitch. Each foot of run takes √(12² + rise²) inches of rafter, the first line of a framing square's rafter table. All of this is geometry and exact.

Roof pitch chart

Whole pitches from 1/12 to 24/12: angle, slope and rafter length per foot of run
PitchAngleSlopeRafter per foot
1/124.76°8.3 %12.042 in
2/129.46°16.7 %12.166 in
3/1214.04°25 %12.369 in
4/1218.43°33.3 %12.649 in
5/1222.62°41.7 %13 in
6/1226.57°50 %13.416 in
7/1230.26°58.3 %13.892 in
8/1233.69°66.7 %14.422 in
9/1236.87°75 %15 in
10/1239.81°83.3 %15.62 in
11/1242.51°91.7 %16.279 in
12/1245°100 %16.971 in
13/1247.29°108.3 %17.692 in
14/1249.4°116.7 %18.439 in
15/1251.34°125 %19.209 in
16/1253.13°133.3 %20 in
17/1254.78°141.7 %20.809 in
18/1256.31°150 %21.633 in
19/1257.72°158.3 %22.472 in
20/1259.04°166.7 %23.324 in
21/1260.26°175 %24.187 in
22/1261.39°183.3 %25.06 in
23/1262.45°191.7 %25.942 in
24/1263.43°200 %26.833 in

From the pitch to the rafter

The rafter's run is half the building's width, measured to the outside of the walls. Its length on the measuring line (the line through the outside corner of the seat cut) is the run times the length per foot, less half the ridge board's thickness along the slope, plus the overhang along the slope. The board is a little longer, by the rafter's depth times the tangent of the angle, because each plumb cut leans across it. The allowed span comes from IRC Tables R802.4.1(1) to (8), which the rafter calculator explains in full.

Framing a deck or a stair as well? The deck calculator sizes joists and beams from the AWC's DCA 6 tables, and the stair calculator lays out a stringer with the same framing-square settings.

Sources

  1. International Code Council, 2024 International Residential Code, Chapter 8: R802.3 Ridge, R802.4.1 and Tables R802.4.1(1)–(9) Rafter spans, R802.6 Bearing, R802.7 Notching; Chapter 5: R502.8.1 (free read-only text), read 2026-11-08
  2. International Code Council, 2021 International Residential Code, Chapter 8: Tables R802.4.1(1)–(9) Rafter spans (free read-only text), read 2026-11-08
  3. NIST, PS 20-25 American Softwood Lumber Standard, Table 3: minimum dressed sizes, read 2026-11-07

How we source these numbers