Radians to Degrees Converter (+ Reference Table)
This converter instantly transforms angles between radians and degrees — the two most important units of angular measurement in mathematics, physics, and engineering. The core formula is degrees = radians × (180 / π) and its inverse radians = degrees × (π / 180). One full revolution equals 2π radians or 360°. Radians are the SI standard unit for angles, preferred in calculus, trigonometry, and physics, while degrees remain the everyday standard in navigation, surveying, and geometry. Use this tool any time you need to switch between unit systems without manual computation.
1 radian = 180/π ≈ 57.2958 degrees. To convert radians to degrees, multiply by 57.2958. To convert degrees to radians, multiply by π/180 ≈ 0.017453. For example: π/2 rad = 90°, π rad = 180°, 2π rad = 360°.
When to use this calculator
- Converting the output of a physics simulation (e.g., angular velocity in rad/s) into degrees for a human-readable display or engineering report.
- Checking trigonometric function inputs when switching between a calculator in degree mode and code that expects radians (e.g., Python's math.sin() uses radians).
- Converting GPS bearing angles (given in degrees) to radians before feeding them into navigation or robotics algorithms.
- Interpreting results from scientific literature — for example, converting a reported phase angle of 1.5708 rad to 90° to confirm it represents a right angle.
- Solving calculus problems where arc length s = rθ requires θ in radians, but the problem states the angle in degrees.
- Converting rotation values in 3D graphics/game engines (e.g., Unity stores rotations in degrees; shaders often need radians).
Worked example: π/4 rad to degrees
- Enter 0.7854 (= π/4) in the angle field
- Select 'Radians → Degrees'
- Result: 45°
How it works
2 min readConversion Formulas
The conversion between radians and degrees is based on the fact that one full circle equals both 2π radians and 360 degrees.
# Radians → Degrees
degrees = radians × (180 / π)
degrees = radians × 57.29577951...
# Degrees → Radians
radians = degrees × (π / 180)
radians = degrees × 0.01745329251...
# Exact constants
π ≈ 3.14159265358979323846
180/π ≈ 57.29577951308232
π/180 ≈ 0.01745329251994330These are exact algebraic relationships — not approximations. The only rounding comes from the precision of π used.
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Reference Table: Common Angles in Degrees and Radians
Memorize the "anchor" values (0, 30, 45, 60, 90, 180, 360) to sanity-check any conversion instantly.
| Degrees (°) | Exact radians | Decimal radians |
|---|---|---|
| 0° | 0 | 0.0000 |
| 15° | π/12 | 0.2618 |
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 120° | 2π/3 | 2.0944 |
| 135° | 3π/4 | 2.3562 |
| 150° | 5π/6 | 2.6180 |
| 180° | π | 3.1416 |
| 270° | 3π/2 | 4.7124 |
| 360° | 2π | 6.2832 |
| 720° | 4π | 12.5664 |
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Worked Examples
Example 1 — Physics problem (rad → degrees):
A pendulum completes an arc of 0.3491 radians. What is that in degrees?
degrees = 0.3491 × (180 / π)
= 0.3491 × 57.2958
= 20.0°Result: 0.3491 rad = 20.0° — a gentle 20-degree swing.
Example 2 — Programming (degrees → rad):
You want to call Math.cos(45°) in JavaScript. JavaScript's Math.cos() expects radians:
radians = 45 × (π / 180)
= 45 × 0.017453
= 0.7854 rad
Math.cos(0.7854) ≈ 0.7071 ✓Example 3 — Navigation bearing:
A drone must turn 270° clockwise. The autopilot algorithm needs radians:
radians = 270 × (π / 180)
= 270 × 0.017453
= 4.7124 rad (= 3π/2)---
Common Mistakes
1. Forgetting that most programming languages use radians by default. Python's math.sin(90) does NOT return 1 — it returns sin(90 rad) ≈ 0.894. You must write math.sin(math.radians(90)) to get 1.0.
2. Multiplying by π instead of dividing (or vice versa). degrees = radians × π gives a result ~17× too large. Always multiply by 57.2958 to go rad→deg, and by 0.017453 to go deg→rad.
3. Treating degrees-minutes-seconds (DMS) as decimal degrees. An angle of 45°30′ is NOT 45.30° — it is 45.5° (30 arc-minutes = 0.5°). Convert DMS to decimal degrees first.
4. Confusing angular velocity with a static angle. When a wheel spins at 10 rad/s, that is angular velocity. Conversion: 10 rad/s × 57.2958 ≈ 572.96 °/s — the factor is the same, but the physical meaning differs.
5. Using an imprecise value of π. Using π ≈ 3.14 instead of 3.14159 introduces ~0.05% error, which compounds in multi-step calculations.
Frequently asked questions
How many degrees is 1 radian?
1 radian equals exactly 180/π degrees, which is approximately 57.2957795°. It seems like an odd number because a radian is defined geometrically: it is the angle subtended at the center of a circle by an arc equal in length to the radius. Since the full circumference is 2πr, there are exactly 2π ≈ 6.2832 radians in a full circle, and 360°/2π ≈ 57.2958° per radian.
What is the formula to convert radians to degrees?
Multiply the angle in radians by (180/π), which is approximately 57.2958. So: degrees = radians × 57.2958. For example, 2 rad × 57.2958 = 114.59°. The reverse is: radians = degrees × (π/180) = degrees × 0.017453.
What are π/6, π/4, π/3, and π/2 in degrees?
These are the most common angles in trigonometry: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°. Also: π = 180° and 2π = 360° (one full revolution). These are worth memorizing for quick mental conversions.
Why do mathematicians and scientists prefer radians over degrees?
Radians make calculus formulas cleaner. For example, the derivative of sin(x) is cos(x) ONLY when x is in radians. In degrees, you'd need an extra factor of π/180. Radians are also the SI standard (NIST SP 811), eliminating conversion overhead in formulas for arc length (s = rθ), angular velocity (ω = θ/t), and rotational kinetic energy.
Do programming languages (Python, JavaScript, C++) use radians or degrees?
Virtually all standard math libraries — Python (math, numpy), JavaScript (Math), C/C++ (math.h), Java (Math), MATLAB, and R — use radians exclusively for all trigonometric functions (sin, cos, tan, asin, etc.). Notable exceptions: Excel and Google Sheets have RADIANS()/DEGREES() helper functions. Always check documentation before assuming the input unit.
How do I convert degrees, minutes, and seconds (DMS) to radians?
First convert DMS to decimal degrees: decimal° = degrees + (minutes/60) + (seconds/3600). Then multiply by π/180. Example: 45°30′18″ → 45 + 30/60 + 18/3600 = 45.505° → 45.505 × 0.017453 = 0.79415 rad. Skipping the DMS→decimal step is the most common error in surveying and GPS work.
Is there a quick mental math trick for radian-to-degree estimation?
Yes — since 1 rad ≈ 57.3°, you can estimate by multiplying radians by 57 for a quick ballpark (good to within ~0.5%). For very small angles (under ~0.3 rad), radians and the sine of the angle are nearly equal (small-angle approximation: sin θ ≈ θ in radians), which is used heavily in optics and structural engineering.
What is a gradian, and how does it compare to degrees and radians?
A gradian (also 'grad' or 'gon') divides a full circle into 400 units, so 1 gradian = 0.9° = π/200 rad. Right angles are exactly 100 gradians, making them convenient in some European surveying systems. Conversion: gradians = degrees × (10/9); gradians = radians × (200/π). Most scientific work uses degrees or radians, not gradians.
Can angles be larger than 2π radians (360°)?
Yes. Angles larger than 2π rad represent multiple full rotations — common in physics and engineering (e.g., a motor spinning at 5 rad/s for 10 seconds accumulates 50 rad ≈ 2,864.8° ≈ 7.96 full rotations). Trig functions are periodic (sin(2π + x) = sin(x)), but the total accumulated angle still carries physical meaning in angular momentum and winding number contexts.