What This Converter Does
The Degrees to Radians Converter handles the angle-unit translation every trig student, calculus student, physicist, programmer, and engineer needs to know cold. Enter an angle in either unit, get the converted value in the other — with a bonus: when the result is a clean fraction of π (like π/2, π/3, 2π/3), the converter shows it in that form too.
60 × (π/180) = π/3 ≈ 1.0472 radians.
Reverse: 1.5708 rad × (180/π) ≈ 90° = π/2 in clean form.
The Two Formulas
Both directions reduce to a single proportional factor:
- Degrees → Radians: multiply by π/180 (≈ 0.01745)
- Radians → Degrees: multiply by 180/π (≈ 57.296)
That's the whole conversion. Memorize that 180° = π radians and you can derive both formulas: 360° = 2π, 90° = π/2, etc.
Common Angles Reference
| Degrees | Radians (decimal) | Radians (π form) | Common context |
|---|---|---|---|
| 0° | 0 | 0 | Starting angle |
| 30° | 0.5236 | π/6 | 30-60-90 triangle |
| 45° | 0.7854 | π/4 | 45-45-90 triangle |
| 60° | 1.0472 | π/3 | 30-60-90 triangle |
| 90° | 1.5708 | π/2 | Right angle |
| 120° | 2.0944 | 2π/3 | — |
| 135° | 2.3562 | 3π/4 | — |
| 180° | 3.1416 | π | Half circle / straight line |
| 270° | 4.7124 | 3π/2 | Three-quarter turn |
| 360° | 6.2832 | 2π | Full circle |
Why Radians Won (At Least in Math)
Degrees are a human convention — 360 was chosen by ancient Babylonian astronomers because it divides cleanly into many factors (2, 3, 4, 5, 6, 8, 9, 10, 12...) and roughly matches the days in a year. They work great for everyday geometry.
Radians are the natural unit of angle in calculus and physics. The reason: sin(x) is a smooth function whose derivative is cos(x) only when x is in radians. In degrees, the derivative would be cos(x) × (π/180) — every formula carries an extra constant. Switching to radians cleans up the math everywhere — Taylor series, Fourier transforms, oscillation equations, the wave equation. By the time you're doing real physics, radians are the default.
Where Each Unit Shows Up
Degrees: everyday geometry, navigation (compass bearings), surveying, architecture, anywhere humans communicate angles. Most CAD tools input angles in degrees.
Radians: calculus, physics, mechanical engineering simulations, signal processing, computer graphics rotations (sometimes), most programming language trig functions.
Both: game development frequently mixes the two — angles displayed to users (and stored in level files) are usually degrees; the actual trig math during runtime uses radians.
The Programming Bug That Bites Everyone Once
Almost every programming language's trig functions take radians, not degrees. Math.sin(90) in JavaScript does NOT return 1 — it returns 0.8939... because it's interpreting 90 as 90 radians (which is many full rotations past 90°). To get sin of 90 degrees:
Math.sin(90 * Math.PI / 180) // = 1, correct
// Or in some languages:
Math.sin(Math.toRadians(90)) // Java equivalentWhenever your trig results look wildly wrong, check the units. Always.
Common Pitfalls
Off-by-π/2. Some applications (especially navigation) measure angle from the y-axis (north) rather than the x-axis (east). Standard math convention is from the x-axis counterclockwise. Mixing conventions gives results 90° off.
Negative angles vs angles > 360°. -45° = 315° = 7π/4 = -π/4 — they're all equivalent. Some functions (atan2) return -π to π; others return 0 to 2π. Know which range you're in before doing math.
Mode setting on calculators. Scientific calculators have a DEG/RAD switch. The most common student-trigonometry mistake is leaving it on the wrong setting. If your sin(30°) doesn't equal 0.5, your calculator is in radian mode.
Related Tools
For triangle geometry where degree-radian conversion comes up, the Pythagorean Theorem calculator handles right triangles. The Distance Formula calculator works in any coordinate system. For circle measurements (where 360° vs 2π comes into play), see the Circumference of a Circle calculator.