Reference Angle Calculator
Find the angle to the nearest x-axis ray and the coterminal angle from 0° to under 360°. Axis cases return 0° or 90° by convention.
Formula
θ₀ = θ − 360° floor(θ/360°); reference = min(θ₀, |180°−θ₀|, 360°−θ₀).
Coterminal angle [0°, 360°) ≈ (225)-360×floor((225)÷360) · Reference angle ≈ min((225),abs(180-(225)),360-(225))
How to use
- Enter the signed angle in degrees.
- Negative angles rotate clockwise.
- Read the reference and the coterminal angle.
Examples
- 225° has reference angle 45°; −30° has reference angle 30°.
Frequently asked questions
What is a reference angle?
For a non-axis terminal ray, it is the acute angle to the x-axis.
What happens on an axis?
This calculator extends the definition: 0° on the x-axis and 90° on the y-axis.
Can I enter more than one turn?
Yes. Whole turns are removed before finding the reference angle.
Sources
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