Introduction
Every number a roof pitch calculator returns — the angle, the slope percentage, the rafter length, the area multiplier — comes from one right triangle: rise (vertical), run (horizontal, always 12 in pitch notation), and the rafter as the hypotenuse. If you understand that triangle, you can reproduce every formula by hand, spot errors in someone else’s numbers, and convert freely between pitch, degrees, and percent slope. This guide derives each formula step by step with worked numbers in imperial and metric.
The Roof Pitch Calculator runs these calculations instantly. What follows is the math behind them, so you can verify any result and use it confidently on site.
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The Core Triangle
Pitch notation like 6/12 defines a right triangle: opposite side (rise) = 6, adjacent side (run) = 12. Everything else is trigonometry on that triangle. The run is a convention — 12 units, any unit — so pitch is a pure ratio: 6/12 = 15 cm / 30 cm = 0.5. That ratio identity is why every formula below is unit-independent; inches and centimeters give the same angle, slope %, and multiplier — the same conversions the FAQ answers at a glance.
Formula 1: Pitch to Degrees
Angle = arctan(rise ÷ 12)
The tangent of the roof angle equals opposite over adjacent — rise over run. So the angle is the inverse tangent (arctangent) of that ratio. For 6/12:
arctan(6 ÷ 12) = arctan(0.5) = 26.57°
More reference points, all computed the same way:
| Pitch | Calculation | Angle |
|---|---|---|
| 3/12 | arctan(0.25) | 14.04° |
| 4/12 | arctan(0.333) | 18.43° |
| 6/12 | arctan(0.5) | 26.57° |
| 8/12 | arctan(0.667) | 33.69° |
| 10/12 | arctan(0.833) | 39.81° |
| 12/12 | arctan(1.0) | 45.00° |
Note the relationship isn’t linear: doubling pitch from 3/12 to 6/12 takes the angle from 14.04° to 26.57°, not double. That’s the arctangent curve at work — a common source of estimating error, discussed in the common mistakes guide.
Formula 2: Degrees to Pitch
Pitch (as x/12) = 12 × tan(angle)
Reading the other way: a speed square or digital level gives you 26.57°, and you need pitch notation. tan(26.57°) = 0.5; 12 × 0.5 = 6 → 6/12. Another: 33.69° → tan = 0.6667 → 12 × 0.6667 = 8 → 8/12. Round to the nearest whole number of twelfths — fractional pitches (9.6/12) are valid but unusual in the field.
Formula 3: Slope Percentage
Slope % = (rise ÷ 12) × 100
This is just the ratio expressed as a percent: 6/12 → 50%, 4/12 → 33.3%, 12/12 → 100%. Percent slope is handy for drainage specs and for communicating with engineers, who often think in percent. Converting back: rise = 12 × (slope % ÷ 100). A 33.3% slope → 12 × 0.333 = 4 → 4/12.
Formula 4: Unit Rafter Length
Unit rafter = √(12² + rise²)
Pythagoras on the core triangle: the rafter is the hypotenuse. For 6/12:
√(144 + 36) = √180 = 13.42 inches of rafter per 12 inches of horizontal run.
To get a full rafter, scale by the actual run: rafter = unit rafter × (actual run in inches ÷ 12), or equivalently rafter = √(run² + rise_total²). A 12 ft run at 6/12: total rise = 12 ft × 0.5 = 6 ft; rafter = √(12² + 6²) = √180 = 13.42 ft. Imperial and metric examples:
| Run | Pitch | Total rise | Rafter length |
|---|---|---|---|
| 12 ft | 6/12 | 6 ft | √(144+36) = 13.42 ft |
| 14 ft | 8/12 | 9.33 ft | √(196+87.1) = 16.83 ft |
| 3.66 m | 6/12 | 1.83 m | √(13.40+3.35) = 4.09 m |
| 4.27 m | 8/12 | 2.85 m | √(18.23+8.10) = 5.13 m |
Remember: the run used here is the horizontal distance from eave to ridge, including any eave overhang if the rafter runs past the wall. The estimating guide shows where overhang enters the quantity build-up.
Formula 5: Roof Area Multiplier
Multiplier = unit rafter ÷ 12, equivalently 1 ÷ cos(angle)
Both forms say the same thing: roof surface per unit of horizontal footprint. For 6/12: 13.42 ÷ 12 = 1.118; check: 1 ÷ cos(26.57°) = 1 ÷ 0.8944 = 1.118. ✓
Why two forms? The rafter form is easier to derive from pitch notation; the cosine form is easier if you measured an angle. Roof area = footprint area × multiplier. A 1,200 sq ft (111.5 m²) footprint at 6/12 → 1,200 × 1.118 = 1,342 sq ft (111.5 × 1.118 = 124.7 m²).
Formula 6: Total Rise from Span
Total rise = (span ÷ 2) × (rise ÷ 12)
For a gable roof, the horizontal run is half the building span. A 28 ft span at 8/12: run = 14 ft; total rise = 14 × (8 ÷ 12) = 14 × 0.6667 = 9.33 ft. Metric: 8.53 m span at 8/12 → run 4.27 m → rise 4.27 × 0.6667 = 2.85 m. Framers use this to set ridge board height; confirm against plans before cutting, then price the resulting quantities with the cost guide.
Worked Verification: Checking Your Numbers
Run the 6/12, 12 ft run example end to end and cross-check:
Step 1 — Angle: arctan(6/12) = 26.57°.
Step 2 — Slope %: 0.5 × 100 = 50%.
Step 3 — Rafter: √(12² + 6²) = 13.42 ft.
Step 4 — Multiplier via rafter: 13.42 ÷ 12 = 1.118.
Step 5 — Multiplier via cosine: 1 ÷ cos(26.57°) = 1.118. Agreement between two independent paths confirms the arithmetic.
Step 6 — Metric: 3.66 m run at 6/12 → rafter √(3.66² + 1.83²) = 4.09 m; 4.09 ÷ 3.66 = 1.118 — same multiplier, as expected.
If your two paths disagree, the usual culprits are measuring the run along the slope instead of horizontally, or rounding the angle before the cosine step. Keep full precision until the final answer.
Related Guides
- Roof Pitch Complete Guide — pitch math, material minimums, and walkability in one place.
- Roof Pitch Estimating Guide — applying these formulas to a real quantity takeoff.
- Pro Tips — field measurement techniques that feed these formulas.
- Common Mistakes — the arithmetic traps this math is meant to catch.
- Roof Pitch FAQ — quick answers on pitch conversions.
- Roof Pitch Cost Guide — how the multiplier turns into dollars.
Get Your Number
Six formulas, one triangle: angle from arctangent, slope % from the ratio, rafter from Pythagoras, multiplier from the rafter or the cosine, total rise from half the span. The Roof Pitch Calculator evaluates them instantly — but now you can verify any result by hand, which is exactly what you want before cutting rafters or ordering material.
Disclaimer: These formulas are geometric planning tools. Real rafters include seat cuts, ridge deductions, and overhang details that change the finished length; material minimums and structural sizing must be verified with the product manufacturer and local building code. Consult a licensed roofer or structural professional for load-bearing and code decisions.
