Radians & Degrees
Convert either way, with the exact π form alongside the decimal.
Exactly π/3
| Degrees | Exact | Decimal |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.523599 |
| 45° | π/4 | 0.785398 |
| 60° | π/3 | 1.047198 |
| 90° | π/2 | 1.570796 |
| 120° | 2π/3 | 2.094395 |
| 135° | 3π/4 | 2.356194 |
| 150° | 5π/6 | 2.617994 |
| 180° | π | 3.141593 |
| 270° | 3π/2 | 4.712389 |
| 360° | 2π | 6.283185 |
A full turn is 360° or 2π radians, so half a turn gives the conversion factor: multiply degrees by π ÷ 180 to get radians, or radians by 180 ÷ π to get degrees. So 60° = π/3 ≈ 1.047198 rad.
Two ways of measuring the same turn
Degrees are a convention. Splitting the circle into 360 parts comes from Babylonian base-60 arithmetic and survives because 360 divides so neatly — by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more, which makes common fractions of a circle come out whole.
Radians are not a convention. One radian is the angle that cuts an arc as long as the radius, which ties the measure to the circle itself rather than to a chosen number of parts. That makes a radian a ratio of two lengths — genuinely dimensionless — which is why it can be dropped from an equation without breaking the units.
Why calculus insists on radians
The derivative of sin x is cos x only when x is in radians. In degrees it picks up a factor of π ÷ 180, because the argument has been rescaled. The same factor infects every series: sin x = x − x³/6 + … is true in radians and false in degrees. So once you reach calculus the choice stops being cosmetic, and every trigonometric identity in an analysis course assumes radians silently.
1 rad ≈ 57.295780°, and 1° ≈ 0.017453 rad
- 1 Decide which way you are going. Degrees to radians multiplies by π ÷ 180; radians to degrees multiplies by 180 ÷ π.
- 2 Write the degrees over 180. For 135°: 135 ÷ 180.
- 3 Reduce the fraction before touching π. 135 ÷ 180 = 3/4, so the answer is 3π/4 — keeping it exact is easier than rounding first.
- 4 Convert to a decimal only if you need one. 3π/4 = 2.356194 radians.
- 5 Going the other way, divide out the π. 5π/6 radians × 180 ÷ π = 5 × 180 ÷ 6 = 150°. The π cancels, which is why exact forms are quicker.
The angles worth memorising
These are the ones that recur on the unit circle and in exam questions.
| Degrees | Exact radians | Decimal |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.523599 |
| 45° | π/4 | 0.785398 |
| 60° | π/3 | 1.047198 |
| 90° | π/2 | 1.570796 |
| 120° | 2π/3 | 2.094395 |
| 135° | 3π/4 | 2.356194 |
| 150° | 5π/6 | 2.617994 |
| 180° | π | 3.141593 |
| 270° | 3π/2 | 4.712389 |
| 360° | 2π | 6.283185 |
Keeping the exact form
Most angles in a trigonometry course are rational multiples of π, and leaving them that way is both more accurate and easier to work with. π/6 and 0.5235987755982988 are the same angle, but only the first makes it obvious that six of them make a half turn. The converter here shows both so you can quote whichever the question wants.
An angle that is not a nice multiple of π simply has no exact form — 1° is π/180, which is exact but not simpler, and 57° is 19π/60. In those cases the decimal is the honest answer, and this page gives it to six places.
One practical trap: a calculator in the wrong mode. If sin(30) returns −0.988 rather than 0.5, it is reading 30 as radians. Every scientific calculator has a DEG/RAD toggle, and most wrong trigonometry answers trace back to it rather than to the arithmetic.