Radian / Degree Calculator
- Multiply the angle in degrees by π/180: 45 × π/180.
- Simplify the fraction: 45/180 = 1/4, so the angle is π/4 radians.
- As a decimal: π/4 ≈ 0.785398 radians.
- Degrees to radians: multiply the angle in degrees by π/180.
- Simplifying: the fraction of π is reduced by dividing the top and bottom by their greatest common factor.
- Full turn: one full turn is 360°, which is 2π radians, so 180° is π radians.
- Gradians: a right angle is 100 gradians, so gradians = degrees × 10/9.
- Converting back: 0.785398 × 180/π = 45°.
- As a share of a full turn, 45° is 12.5% of 360°.
- In gradians the same angle is 50, where a right angle is 100.
Radian / Degree Calculator: Convert Angles Both Ways
This free radian to degree calculator converts an angle between degrees and radians in either direction. It gives the exact answer in terms of π when there is a simple one, the decimal value, the angle in gradians and turns, and in degrees, minutes and seconds. A unit circle shows where the angle points, and the working is written out so you can follow it.
Quick answer: 45° equals π/4 radians, which is about 0.7854. To go the other way, π/4 radians equals 45°.
Degrees and Radians
Degrees split a full turn into 360 equal parts, a choice that goes back to ancient astronomy and works well for everyday measuring. Radians measure an angle by the arc it cuts off on a circle with a radius of 1. Because the whole circle has a length of 2π, a full turn is 2π radians, and 180° is exactly π radians. Radians are the natural unit in calculus, physics and programming, where formulas for derivatives and waves are simplest when angles are in radians.
How to Convert Degrees to Radians
Multiply the degrees by π and divide by 180. For 60°, that is 60π/180, which simplifies to π/3. The calculator reduces the fraction for you and also gives the decimal. This is the same step used with the sin cos tan calculator, where the Radians button reads the angle in this unit.
How to Convert Radians to Degrees
Multiply the radians by 180 and divide by π. If the angle is written with π, the π cancels, so 3π/2 radians becomes 3 × 180 ÷ 2 = 270°. A plain decimal such as 2 radians is multiplied by 180 and divided by 3.14159, giving about 114.59°. You can type angles with pi in the radians box, for example pi/6 or 5pi/12.
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 180° | π | 3.1416 |
| 270° | 3π/2 | 4.7124 |
| 360° | 2π | 6.2832 |
The Unit Circle
On the unit circle, an angle is measured from the positive x-axis, counter-clockwise for positive values and clockwise for negative ones. The graph above shades the angle you entered and marks the quarter turns at 0, π/2, π and 3π/2. Angles bigger than one full turn go around again, so 405° points the same way as 45°. The calculator tells you when this happens.
Gradians, Turns and Minutes
Besides radians, angles are measured in gradians, where a right angle is 100, and in turns, where one turn is a full rotation. Land surveying often uses degrees, minutes and seconds, in which a degree is split into 60 minutes and each minute into 60 seconds. All of these are shown below the main answer. For the next steps with angles, the law of sines calculator solves triangles, and the circle calculator, the scientific calculator and the math solver cover related work.
How to Use This Radian / Degree Calculator
Choose the direction
Pick Degrees to radians or Radians to degrees.
Enter the angle
Type it, drag the slider, or tap a common angle. Radians can be typed with pi.
Read the answer
See the exact value, the decimal, gradians, turns and degrees, minutes and seconds.
Check the circle
Watch the angle move on the unit circle and follow the working.