What is a radian?
A radian is an angle measurement based on a circle’s radius, not on dividing a circle into 360 equal parts. One radian is the angle created when the arc length equals the radius. Because of this geometric definition, radians are the natural unit for trigonometry, calculus, and many physics formulas.
Degrees and radians: quick formulas
Degrees to radians
radians = degrees × (π / 180)
Radians to degrees
degrees = radians × (180 / π)
This calculator supports both directions, so it works as a degrees to radians calculator and a radians to degrees calculator.
For radian input, you can type values like pi/6, 3*pi/4, or 2.5.
Common angle conversions
- 30° = π/6
- 45° = π/4
- 60° = π/3
- 90° = π/2
- 180° = π
- 270° = 3π/2
- 360° = 2π
Why radians matter in math and science
1) Trigonometry and the unit circle
Standard trigonometric identities are most elegant in radians. On the unit circle, an angle in radians directly tells you how much arc length has been covered.
2) Calculus
The derivative rules for sine and cosine are clean only in radians:
d/dx (sin x) = cos x and d/dx (cos x) = -sin x.
If you use degrees, extra conversion factors appear.
3) Physics and engineering
Angular velocity, oscillations, waves, and rotational motion formulas often assume radian measure. Using radians prevents unit mistakes and keeps equations consistent.
How to use this radian calculator
- Select conversion mode.
- Enter your angle value.
- Choose decimal precision.
- Press Calculate.
Tip: In radians mode, you can type pi or the symbol π.
Example inputs: pi/3, 2*pi, 1.0472.
FAQ
Is 1 radian equal to 57.3 degrees?
Approximately, yes. More precisely: 1 rad ≈ 57.2958°.
How many radians are in a full circle?
A full turn is 2π radians, which is also 360°.
Should I learn both units?
Yes. Degrees are intuitive for everyday angles, while radians are essential for advanced math, coding, and science.