Convert an angle either way and see the exact answer as a multiple of π alongside the decimal, plus the standard-angle table.
A full turn is 360°, and it is also 2π radians. Everything follows from that single equality:
radians = degrees × π / 180 · degrees = radians × 180 / π
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. That is why it is the natural unit for calculus: the derivative of sin x is cos x only when x is in radians. In degrees an awkward factor of π/180 appears and never goes away.
| Degrees | Radians (exact) | Decimal |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 120° | 2π/3 | 2.0944 |
| 135° | 3π/4 | 2.3562 |
| 180° | π | 3.1416 |
| 270° | 3π/2 | 4.7124 |
| 360° | 2π | 6.2832 |
Cancel before multiplying out. Leaving the answer as a fraction of π is almost always what the mark scheme wants.
Multiply the angle in degrees by pi and divide by 180. So 60 degrees becomes 60 pi over 180, which simplifies to pi over 3.
Multiply the angle in radians by 180 and divide by pi. So pi over 4 radians becomes 180 over 4, which is 45 degrees.
One radian is 180 divided by pi degrees, about 57.2958 degrees. It is the angle subtended at the centre of a circle by an arc whose length equals the radius.
Because the derivative of sin x equals cos x only when x is in radians. Working in degrees introduces a constant factor of pi over 180 into every derivative and integral of a trigonometric function.
Two pi radians, approximately 6.2832. A half turn is pi radians, or 180 degrees.