Degrees to Radians Converter
Convert degrees to radians instantly. Essential for mathematics, physics, engineering, and programming.
Last updated: May 2026
Enter a value to see the conversion instantly.
How to convert
Enter a degree value in the "From" field to instantly see the equivalent in radians. The conversion uses the formula: radians = degrees × (π / 180). You can use the preset buttons (0°, 45°, 90°, 180°, 360°) to quickly test common angles, or swap the units to convert radians back to degrees.
Common reference values
| Degrees | Radians | Context |
|---|---|---|
| 0° | 0 | Starting angle (0 radians) |
| 30° | π/6 ≈ 0.5236 | One-twelfth of a full circle |
| 45° | π/4 ≈ 0.7854 | One-eighth of a full circle |
| 60° | π/3 ≈ 1.0472 | Equilateral triangle angle |
| 90° | π/2 ≈ 1.5708 | Right angle (quarter circle) |
| 180° | π ≈ 3.1416 | Straight angle (half circle) |
| 360° | 2π ≈ 6.2832 | Full circle (complete rotation) |
Understanding Degrees and Radians
Both degrees and radians measure angles, but they use different scales. Degrees divide a full circle into 360 equal parts. Radians relate the angle to the arc length on a unit circle—a full circle is 2π radians. The conversion relationship is: 360° = 2π radians, or equivalently, 180° = π radians.
Why Radians Matter
Radians are the natural choice for: Calculus and trigonometric derivatives (derivatives of sine and cosine are simpler in radians), physics equations involving circular motion and oscillation, engineering calculations with rotational systems, and most programming languages (trigonometric functions like sin() and cos() expect radians by default).
Degrees are preferred for: Navigation and surveying (compass bearings), everyday angle descriptions (like camera field of view), and any context where fractional parts of angles are common.
Practical Context
- 45° / 0.785 rad: Optimal angle for projectile range, typical camera tilt
- 90° / 1.571 rad: Right angle, perpendicular lines, quarter turn
- 180° / 3.142 rad: Straight angle, complete reversal, π radians
- 360° / 6.283 rad: Full rotation, one complete circle, 2π radians
Frequently Asked Questions
Why are radians used in mathematics and programming?
Radians are used because they have a direct geometric relationship to the circle itself: the radian measure equals the arc length on a unit circle. This makes calculus much cleaner—the derivatives of sine and cosine are simpler and more elegant in radians. Most programming languages use radians for trigonometric functions (sin, cos, tan) because of these mathematical properties.
What is π (pi) in the conversion?
π (pi) is approximately 3.14159... and represents the ratio of a circle's circumference to its diameter. In angle conversion, π radians equals exactly 180°. Therefore, to convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
Can I convert decimal degrees?
Yes. The converter handles any decimal value. For example, 22.5° = π/8 ≈ 0.3927 radians, and 37.5° ≈ 0.6545 radians. Decimal degrees are common in GPS coordinates, surveying, and scientific measurements.
Why does my calculator give different results?
Make sure your calculator is set to the correct mode: if you're working with degrees, use degree mode; if you're working with radians, use radian mode. A common mistake is entering degrees but having the calculator in radian mode, or vice versa. This converter always shows the conversion between the two systems.
What's the relationship between radians and trigonometry?
In radians, many trigonometric identities simplify beautifully. For instance, sin(x) ≈ x for small values of x (in radians), and the Taylor series for sine, cosine, and exponential functions are most elegant when angles are in radians. This is why scientific calculators and programming languages default to radians.