Radians ↔ Degrees Converter
Convert radians to degrees (and back) instantly. Supports π inputs like π/6 and shows both decimal and π-fraction forms (when possible).
Quick rule to remember
Degrees → Radians: multiply by π/180
Radians → Degrees: multiply by 180/π
Background
Degrees are common in geometry, while radians are common in calculus and trig identities. This converter helps you switch units fast — plus it recognizes the “nice angles” on the unit circle.
How to use this converter
- Choose a mode: Radians → Degrees, Degrees → Radians, or Show both.
- Enter your value (supports π like 3π/4), then click Calculate.
- Use Quick picks for common unit-circle angles.
- Optional: turn on Normalize angle to see an equivalent angle in a standard range like [0°, 360°) or [−π, π).
How this converter works
- It converts using the exact relationship π rad = 180°.
- If your radian result is close to a “nice” multiple of π, the converter can show the π-fraction form.
- If normalization is enabled, it adds or subtracts full turns (360° or 2π) to land in the chosen range.
- It also recognizes common unit-circle angles, and shows the quadrant and reference angle.
Formula & Equation Used
Radians → Degrees: degrees = radians × (180/π)
Degrees → Radians: radians = degrees × (π/180)
Normalization idea: θ ≡ θ + 360°k or θ ≡ θ + 2πk for any integer k
Example Problem & Step-by-Step Solution
Example 1 — Convert 3π/4 to degrees
- Use degrees = radians × (180/π).
- (3π/4) × (180/π) = (3/4) × 180 = 135°.
- It’s a unit-circle angle in Quadrant II with reference angle 45°.
Example 2 — Convert 210° to radians (and normalize)
- Use radians = degrees × (π/180).
- 210 × (π/180) = (210/180)π = (7/6)π.
- So 210° = 7π/6 rad.
- Unit-circle insight: 210° is in Quadrant III with reference angle 30°.
- If you normalize to [−π, π): 7π/6 − 2π = −5π/6.
Example 3 — Convert −5π/6 to degrees (and normalize)
- Use degrees = radians × (180/π).
- (−5π/6) × (180/π) = −5 × 30 = −150°.
- So −5π/6 = −150°.
- If you normalize to [0°, 360°): −150° + 360° = 210°.
- Unit-circle insight: −150° (or 210°) is Quadrant III, reference angle 30°.
Frequently Asked Questions
Q: Why do radians show up so much in calculus?
Because many trig formulas (especially derivatives/integrals) work cleanly only when angles are measured in radians.
Q: What does “reference angle” mean?
It’s the acute angle between your terminal side and the x-axis. It helps you use unit-circle trig values quickly.
Q: Does normalization change my input?
No — it only shows an equivalent angle that lands in a standard range like [0°, 360°) or [−π, π).
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