Calcylator
Roof pitch angle

Roof pitch angle:
formula and worked example

Take the arctangent of rise over run to turn a pitch ratio such as 3:12 into the angle you set on a saw or a roofing guide.

Calcylator Editorial Team

Updated · 4 min read

Rise, run and the angle they describe

Roof slope is described in several ways: a ratio such as 3:12, a percentage, or an angle in degrees. Carpenters work in ratios because they are easy to mark with a square, while manufacturers of tiles, shingles and metal sheet often state minimum angles in degrees.

The two quantities are linked by the right triangle formed by the rafter. Rise is the vertical gain, run is the horizontal distance over which that gain occurs, and the angle at the eave follows from them.

Why does this matter? Pitch controls how quickly water leaves the roof, how much of the attic is usable, how wind loads act on the surface, and what materials may legally and practically be laid. It also drives cost, because a steeper roof has more surface for the same footprint and often needs additional safety measures during construction.

The formula and its inputs

Roof pitch angle =degrees(atan(rise ÷ run))
rise:
Rise
run:
Run
Rise and run must be in the same unit, but the unit itself does not matter. The calculator must be in degree mode.
  • Rise: the vertical height gained by the roof surface, in any consistent length unit.
  • Run: the horizontal distance over which that rise happens; in a symmetrical gable it is half the building span, measured from the wall plate to the ridge line.

The ratio rise ÷ run is the tangent of the angle, so the arctangent (atan) returns the angle itself.

Worked example for a 3-in-12 roof

A roof rises 3 units for every 12 units of run, a common ratio on low-slope residential roofs. Illustrative values.

  • Rise

    3

  • Run

    12

Roof pitch angle

14.036 degrees

Worked arithmetic from the displayed inputs; real jobs will differ with site conditions and measurement error.

  1. Form the ratio: 3 ÷ 12 = 0.25.
  2. Take the arctangent: atan(0.25) ≈ 0.2450 radians.
  3. Convert radians to degrees: 0.2450 × 180 ÷ π ≈ 14.036 degrees.
  4. As a percentage slope, the same roof is 25%.

Reading 14 degrees on a real roof

Different roofing materials have minimum slopes for water shedding, and these depend on the product, the underlay and the local climate, so the manufacturer's installation guide governs. As general conventions, very low slopes need continuous membranes, while steeper slopes shed rain, leaves and snow more reliably but cost more in material and scaffolding.

Angle also affects the rafter length. For a 3:12 roof, the rafter is about 1.03 times the run, because sqrt(1 + 0.25²) ≈ 1.031. At steeper angles, that factor climbs quickly, so a small change in slope can add noticeably to roofing area and cost.

The usable attic space improves with slope, but not evenly. At low slopes there is little headroom, and every extra degree creates only a modest gain. At moderate and steep angles the loft becomes a genuine room. If you plan to use the roof void for storage or living space, test headroom early with a section drawing at the pitch you are considering.

Why a 10% steeper rise is not 10% more degrees

The tangent function is not linear, so percentage changes in the ratio do not map to equal percentage changes in the angle.

Effect of a 10% change: roof pitch angle
ScenarioRiseCalculated result
Original assumptions314.036 degrees
Rise increased by 10%3.315.376 degrees

Raising the rise from 3 to 3.3 increases the ratio by 10% but the angle only goes from 14.036 to 15.376 degrees, an increase of about 9.5%. At steeper pitches the gap widens. For quick checks, remember that near 14 degrees one degree of angle is worth only about 0.22 of a unit of rise per 12 units of run.

Slips that give the wrong pitch

  • Using the full building span as run in a symmetrical gable, which halves the apparent slope.
  • Measuring rise along the sloping rafter instead of straight up, which makes the ratio look steeper than it is.
  • Leaving a calculator in radian mode and reading 0.245 as degrees.
  • Mixing units, such as rise in millimetres and run in metres.

On-site check of a roof slope

  1. Measure run horizontally from the point directly below the ridge to the outer edge of the wall plate.
  2. Measure rise vertically from the plate level to the ridge or to the point on the rafter.
  3. Divide and take the arctangent in degree mode.
  4. Compare the angle with the manufacturer's minimum for the roofing material.

If the angle falls below the minimum, the options are a steeper frame or a different roofing system, and that decision is easiest to make before the rafters are cut.

A site tip is to use a speed square or an inclinometer app and compare the reading with the formula. If the roof is already built, you can measure rise and run on a rafter with a spirit level and tape, which avoids climbing to the ridge. Take the reading at two rafters to confirm the framing is consistent.

Common questions

What angle is a 3-in-12 roof pitch?

A 3-in-12 pitch rises 3 units over 12 units of run, which equals atan(3 ÷ 12), about 14.04 degrees. As a percentage slope it is 25%. The same angle applies whether you measure in inches, centimetres or metres.

How do I convert roof pitch to degrees?

Divide rise by run, then take the inverse tangent of that ratio with your calculator in degree mode. For example, 6 over 12 gives atan(0.5), roughly 26.57 degrees. Always confirm that rise and run use the same unit.

Is roof pitch the same as roof slope?

They are often used interchangeably, but pitch in some regions means rise over the full span, while slope means rise over run. Confirm which definition a plan or supplier uses, as the numbers differ by a factor of two.

Why is run half the span on a gable roof?

The ridge sits above the centre of the building, so each roof plane covers only half the span horizontally. Rise is measured up to the ridge over that half span, so the run for each side equals half the total span.

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