Degrees to Radians Converter
Convert any angle between degrees and radians. Enter a value in either field and get the converted result with the full calculation shown.
The Conversion Formulas
Degrees and radians both measure angles, just on different scales. A full rotation is 360° or $2\pi$ radians. This gives the two conversion formulas:
$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$ $$\text{degrees} = \text{radians} \times \frac{180}{\pi}$$Example: 45° to Radians
Multiply by $\dfrac{\pi}{180}$:
$$45 \times \frac{\pi}{180} = \frac{45\pi}{180} = \frac{\pi}{4} \approx 0.7854 \text{ rad}$$Example: $\dfrac{3\pi}{2}$ Radians to Degrees
Multiply by $\dfrac{180}{\pi}$:
$$\frac{3\pi}{2} \times \frac{180}{\pi} = \frac{3 \times 180}{2} = 270°$$Common Angles Reference Table
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 0° | $0$ | 0 |
| 30° | $\dfrac{\pi}{6}$ | ≈ 0.5236 |
| 45° | $\dfrac{\pi}{4}$ | ≈ 0.7854 |
| 60° | $\dfrac{\pi}{3}$ | ≈ 1.0472 |
| 90° | $\dfrac{\pi}{2}$ | ≈ 1.5708 |
| 120° | $\dfrac{2\pi}{3}$ | ≈ 2.0944 |
| 135° | $\dfrac{3\pi}{4}$ | ≈ 2.3562 |
| 150° | $\dfrac{5\pi}{6}$ | ≈ 2.6180 |
| 180° | $\pi$ | ≈ 3.1416 |
| 270° | $\dfrac{3\pi}{2}$ | ≈ 4.7124 |
| 360° | $2\pi$ | ≈ 6.2832 |
What is a Radian?
A radian is the angle subtended at the center of a circle by an arc whose length equals the radius of that circle. Because the circumference of a circle is $2\pi r$, a full rotation contains $2\pi$ radians — approximately 6.2832 rad.
Most scientific calculators default to radians. If you ever get an unexpected result when computing sin(90), for example, check whether your calculator is in degree or radian mode — $\sin(90°) = 1$ but $\sin(90\text{ rad}) \approx 0.894$.
Frequently Asked Questions
Why do mathematicians prefer radians?
Radians simplify calculus formulas. The derivative of $\sin(x)$ is $\cos(x)$ only when $x$ is in radians. In degrees, an extra $\pi/180$ factor appears everywhere, making formulas messier. Radians are the natural unit of angle measurement in mathematics.
How do I convert a radian expressed as a multiple of π?
If you have something like $3\pi/4$, substitute the decimal value of $\pi \approx 3.14159$ to get $3 \times 3.14159 / 4 \approx 2.356$ rad. Or multiply the coefficient of $\pi$ by 180 to convert directly to degrees: $3/4 \times 180 = 135°$.
Can I use this converter for negative angles?
Yes — negative angles represent clockwise rotation. The same formulas apply: $-90° \times \pi/180 = -\pi/2 \approx -1.5708$ rad.
What are gradians?
Gradians (also called gon or grad) divide a full circle into 400 units instead of 360. They are occasionally used in surveying. To convert: $1° = 10/9$ grad; $1 \text{ grad} = 0.9°$. Most trig courses only require degrees and radians.
