Degrees to Radians Converter
Enter an angle in degrees and this tool returns the exact value in radians, as a clean π fraction when possible, and as a decimal always.
Degrees to radians formula
radians = degrees × (π ÷ 180)
Because this is just multiplication by a constant, the conversion scales linearly. Doubling the degree value doubles the radian value, and halving one halves the other. That linear relationship is what makes a single reference table, like the one further down this page, cover every commonly used angle without needing separate rules for different ranges.
Why π ÷ 180? A full circle is 360° and also 2π radians. Dividing both sides by 360 gives 1° = π/180 radians, the constant that converts any degree measure into radians by simple multiplication.
Radians are used throughout trigonometry, calculus, and physics because they measure angles as a ratio of arc length to radius, a pure number with no arbitrary unit attached. This makes derivatives and integrals of trigonometric functions work out cleanly, which degrees cannot do. A reference angle expressed in radians follows the exact same quadrant rules as one in degrees, just with π substituted for 180° and 2π substituted for 360°.
A radian is defined directly from the circle itself: it's the angle created when the arc length along the circle equals the radius. Walk a distance equal to one radius along the edge of a circle, then draw a line from that point back to the center. The angle at the center is exactly 1 radian, about 57.3°. Degrees, by contrast, are an arbitrary human invention. The 360° convention traces back to ancient Babylonian astronomy, likely chosen because 360 divides evenly by so many small numbers (2, 3, 4, 5, 6, 8, 9, 10, 12…), which made it convenient for hand calculation long before calculators existed.
Radians matter beyond pure math too. The arc length of a circle is given by s = rθ, but only when θ is measured in radians. Plug in degrees and the formula breaks. Angular velocity in physics, measured in radians per second, describes how fast something rotates without referencing an arbitrary degree scale. Calculus depends on radians even more directly: the derivative of sin(x) is cos(x) only when x is in radians. Using degrees introduces an extra conversion constant into every derivative and integral, which is why radians become the default unit the moment trigonometry meets calculus.
None of this changes how reference angles work. Whether an angle is written as 210° or 7π/6, the underlying geometry, the terminal side, the quadrant, the acute distance to the x-axis, is identical. Converting units first is often just a convenience, chosen because the resulting numbers are easier to work with in a particular formula or field.
Students often ask whether they should "think in degrees" or "think in radians." The honest answer is both, depending on context. Geometry problems, navigation, and everyday angle measurements, the slope of a roof, the swing of a door, are almost always described in degrees, because whole numbers like 30°, 45°, and 90° are easy to picture and communicate. Calculus, physics simulations, and computer graphics almost always work in radians internally, because the mathematics simplifies dramatically once the arbitrary 360° convention is removed. Learning to convert fluently in both directions means never being stuck translating a problem from the notation it was given in into the notation you're comfortable working with.
Step-by-step conversion guide
Worked example: 90°
- Start with the degree value: 90°
- Multiply by π ÷ 180: 90 × π/180 = 90π/180
- Simplify the fraction: 90π/180 = π/2
90° = π/2 radians ≈ 1.5708
Worked example: 200°
- Start with the degree value: 200°
- Multiply by π ÷ 180: 200π/180
- Simplify: 200π/180 = 10π/9
200° = 10π/9 radians ≈ 3.4907
Not every degree value simplifies to a clean π fraction. Values that don't divide evenly into 180, most decimal degree measurements, for instance, produce an irrational radian value with no exact fraction at all. In that case, the decimal approximation is the only reasonable way to express the result, which is exactly what this calculator falls back to automatically.
A quick sanity check for any conversion: 1° is a small slice of a full circle, so it should convert to a small radian value, about 0.0175. If a result comes out larger than roughly 6.28 (2π radians, a full turn), the degree value entered was probably larger than 360° to begin with, or a step in the arithmetic was skipped.
Common degree-to-radian conversions
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 0° | 0 | 0.0000 |
| 15° | π/12 | 0.2618 |
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 75° | 5π/12 | 1.3090 |
| 90° | π/2 | 1.5708 |
| 105° | 7π/12 | 1.8326 |
| 120° | 2π/3 | 2.0944 |
| 135° | 3π/4 | 2.3562 |
| 150° | 5π/6 | 2.6180 |
| 165° | 11π/12 | 2.8798 |
| 180° | π | 3.1416 |
| 195° | 13π/12 | 3.4034 |
| 210° | 7π/6 | 3.6652 |
| 225° | 5π/4 | 3.9270 |
| 240° | 4π/3 | 4.1888 |
| 270° | 3π/2 | 4.7124 |
| 300° | 5π/3 | 5.2360 |
| 315° | 7π/4 | 5.4978 |
| 330° | 11π/6 | 5.7596 |
| 360° | 2π | 6.2832 |
Frequently Asked Questions
How do you convert degrees to radians?
Multiply the degree value by π ÷ 180. For example, 90° × (π ÷ 180) = π/2 radians. This formula works for any angle, including negative angles and angles larger than 360°, since it is just a proportional scaling factor between the two units.
What is 180 degrees in radians?
180° equals π radians. This is the definition that connects the two units: a straight angle (half a full turn) is π radians. Every other degree-to-radian conversion follows from this single relationship, since 1° always equals π ÷ 180 radians.
What is 90 degrees in radians?
90° equals π/2 radians, since 90 × (π ÷ 180) = π/2. This is one of the five special angles, along with 0°, 30°, 45°, and 60°, whose radian equivalents are worth memorizing directly rather than calculating each time they come up.
What is 45 degrees in radians?
45° equals π/4 radians, since 45 × (π ÷ 180) = π/4. Because 45° is exactly half of 90°, its radian value is also exactly half of π/2, which makes π/4 easy to recall from the quadrantal angle directly above it.
What is 360 degrees in radians?
360° equals 2π radians, a full turn around the circle. Any angle larger than 360° can be reduced first by subtracting multiples of 2π radians, the same way multiples of 360° are subtracted in degrees, before applying the standard quadrant formula.
Does this tool accept gradians or arc-minutes?
This calculator accepts degrees and radians, the two units used in almost all trigonometry coursework, but it does not accept gradians or arc-minutes directly. To use a gradian value here, multiply it by 360 divided by 400 to convert to degrees first. For arc-minutes, divide by 60, and for arc-seconds, divide by 3600.