Radians to Degrees Converter
Convert an angle either direction — radians to degrees or degrees to radians — using π.
The conversion formulas
Angles come in two common flavours. Degrees split a full turn into 360 equal parts — a convention we inherited from Babylonian astronomers. Radians measure the same angle by the arc it sweeps out on a unit circle, so a full turn is 2π radians. To move between them:
- Radians → degrees:
degrees = radians × 180 / π - Degrees → radians:
radians = degrees × π / 180
The two formulas are mirror images: one multiplies by 180/π, the other by its reciprocal π/180.
Why π shows up
Walk exactly halfway around a circle and you've turned through a straight line — 180°. In radians, that same half-turn is the length of half the circumference of a unit circle, which is π. So the fundamental fact behind every conversion here is:
π radians = 180°
Everything else is just scaling that identity. Divide both sides by 180 and you get the degrees-to-radians factor; divide by π and you get the radians-to-degrees factor. Pi (π) is an irrational number — its decimal expansion runs on forever without repeating — which is why a "nice" angle in degrees like 60° turns into the slightly awkward-looking 1.047… radians. It's cleaner to write it as π/3.
Quick reference table
| Degrees | Radians |
|---|---|
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 180° | π |
| 360° | 2π |
These six are worth committing to memory — most trigonometry problems lean on them. Notice the pattern: as the degrees climb, the radian value climbs in tidy fractions of π.
Worked examples
Radians to degrees. Convert π/2 radians. As a decimal that's about 1.5708. Multiply by 180/π:
1.5708 × 180 / π ≈ 90°
Degrees to radians. Convert 60°. Multiply by π/180:
60 × π / 180 = π/3 ≈ 1.047198 rad
The converter above shows this second kind of answer as a clean multiple of π whenever one exists, so you'll see π/3 rather than just a run of decimals.
When to use radians
Degrees are friendlier for everyday geometry, navigation, and anything you'd describe out loud ("turn 90 degrees left"). Radians are the natural choice in calculus, physics, and programming: the derivative rules for sine and cosine only come out clean when angles are in radians, and functions like Math.sin() in most languages expect radians, not degrees. If a trig formula misbehaves in your code, a forgotten degree-to-radian conversion is the usual culprit.
Related
- Reference angle calculator — fold any angle back into the first quadrant.
- Arc length calculator — where radians earn their keep.
- What is pi? — the constant behind these formulas.
- Digits of pi — how far π's decimals really go.