Inverse Tangent (Arctan) Calculator
Calculate angles from slope ratios or opposite/adjacent sides. Features a specialized 'Slope Percentage to Angle' converter for roofing and grading.
About
The Inverse Tangent function, arctan(x), relates the slope of a line to its angle. In a right-angled triangle, it is the angle whose tangent is the ratio of the opposite side (rise) to the adjacent side (run).
This tool is indispensable in construction for determining roof pitches, road gradients, and ramp slopes. Unlike sine and cosine which are bound between -1 and 1, the tangent function can take any real number as input, from negative infinity to positive infinity, returning an angle between -90° and 90°.
arctan calculator
inverse tangent
slope calculator
roof pitch
Formulas
From Sides:
θ = arctan(OppositeAdjacent)
From Slope Percentage:
θ = arctan(Slope% / 100)
Reference Data
| Slope % | Ratio (Rise/Run) | Angle (Degrees) | Common Use |
|---|---|---|---|
| 1.7% | 1:60 | 1° | Drainage Min. |
| 5% | 1:20 | 2.9° | Road Max (Hwy) |
| 8.3% | 1:12 | 4.8° | Wheelchair Ramp |
| 100% | 1:1 | 45° | Steep Roof (12/12) |
| 173% | √3 | 60° | Geometry |
| ∞ | 1:0 | 90° | Vertical Wall |
Frequently Asked Questions
Arctan(y/x) only returns angles in quadrants 1 and 4 (-90° to 90°). It loses information about which specific quadrant the point lies in (e.g., it can't distinguish between Quadrant 1 and 3). Arctan2(y, x) takes both coordinates separately and returns the correct angle across all 360° (-180° to 180°).
A '6/12' pitch means a rise of 6 for every 12 of run. Divide 6 by 12 to get 0.5. Then take the arctan(0.5) to get approximately 26.6°.
A 100% slope means the Rise equals the Run (1/1 ratio). This results in a 45-degree angle. It is a common misconception that 100% slope means a vertical 90-degree wall.