Degrees to Radians Converter
Convert degrees to radians and radians to degrees, with the exact answer as a fraction of π (90° = π/2) plus turns and gradians.
radians = degrees × π/180
The Formula: Multiply by π/180
A full circle is 360° or 2π radians, so 180° = π radians. Multiply degrees by π/180 to get radians, and multiply radians by 180/π to get degrees. One radian is the angle at which the arc around a circle is exactly as long as the radius: about 57.2958°.
For an exact answer, keep π as a symbol and simplify the fraction: 135° × π/180 = 135π/180 = 3π/4.
Unit Circle Table: Degrees, Radians, Sin and Cos
The angles that come up most in trigonometry, with exact radian values and their sine and cosine.
| Degrees | Radians (exact) | Radians (decimal) | sin | cos |
|---|---|---|---|---|
| 0° | 0 | 0 | 0 | 1 |
| 30° | π/6 | 0.5236 | 1/2 | √3/2 |
| 45° | π/4 | 0.7854 | √2/2 | √2/2 |
| 60° | π/3 | 1.0472 | √3/2 | 1/2 |
| 90° | π/2 | 1.5708 | 1 | 0 |
| 120° | 2π/3 | 2.0944 | √3/2 | −1/2 |
| 135° | 3π/4 | 2.3562 | √2/2 | −√2/2 |
| 150° | 5π/6 | 2.618 | 1/2 | −√3/2 |
| 180° | π | 3.1416 | 0 | −1 |
| 270° | 3π/2 | 4.7124 | −1 | 0 |
| 360° | 2π | 6.2832 | 0 | 1 |
Worked Example: Why Math.sin(90) Isn't 1
You expect the sine of a right angle to be 1, but Math.sin(90) in JavaScript returns 0.894. Nearly every programming language's trig functions take radians, so the code calculated the sine of 90 radians, which is more than 14 full turns.
Step 1: Convert the angle first
Step 2: Pass radians to the function
| Language | Correct call |
|---|---|
| JavaScript | Math.sin(90 * Math.PI / 180) |
| Python | math.sin(math.radians(90)) |
| Excel / Sheets | =SIN(RADIANS(90)) |
| C / C++ | sin(90 * M_PI / 180.0) |
Spreadsheets work the same way: =SIN(90) gives 0.894, and =SIN(RADIANS(90)) gives 1.
Worked Example: Arc Length From an Angle
A bicycle wheel with a 0.35 m radius turns through 120°. How far does a point on the tyre travel? Arc length is s = r × θ, and it only works with θ in radians.
Step 1: Convert to radians
Step 2: Multiply by the radius
The tyre rolls 0.733 m. Using 120 directly would give 42 m, which is why radians are called the "natural" unit: they link angle, radius and distance with no extra constant.
Degrees, Minutes and Seconds to Radians
Maps, GPS coordinates and surveying often write angles as degrees, minutes and seconds (DMS). A degree has 60 minutes (′) and a minute has 60 seconds (″). Convert to decimal degrees first, then to radians:
Turns and Gradians
The converter also shows each angle as a fraction of a turn and in gradians, two other units you may see in software and surveying.
| Unit | Full circle | Right angle | Used in |
|---|---|---|---|
| Turn (revolution) | 1 | 0.25 | Animation, CSS (turn), rotation counts |
| Degree | 360 | 90 | Everyday angles, navigation, construction |
| Radian | 2π ≈ 6.2832 | π/2 ≈ 1.5708 | Maths, physics, programming |
| Gradian (gon) | 400 | 100 | Surveying, some European calculators |
Frequently asked questions
How many degrees is 1 radian?
About 57.2958°, which is 180/π. It's an irrational number, so radians and degrees never convert to each other as whole numbers except at 0.
How many radians are in a full circle?
2π, about 6.2832. Half a circle is π (180°) and a right angle is π/2 (90°).
Why do radian answers usually include π?
Because 180° is exactly π radians, any whole number of degrees is an exact fraction of π. Writing 3π/4 is exact, while 2.3562 is a rounded decimal of the same angle.
Can angles be negative or larger than 360°?
Yes. Negative angles turn clockwise, and angles over 360° include full extra turns: 450° is 5π/2, one turn plus a right angle. The converter accepts both.
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