Angle Converter
A full turn is 360 degrees, 2π radians or 400 gradians. So 180° is π radians, about 3.1416, and 90° is 100 gradians. Radians are the mathematical default because trigonometric functions in every programming language expect them — passing degrees to Math.sin is a classic silent bug.
Enter a value, pick the units, and get an instant result — plus a live table showing that same angle in all three units at once. Nothing you enter ever leaves your browser.
| Degrees | Radians | Gradians |
|---|---|---|
| — | — | — |
Degrees, radians, and gradians — one pivot, three units
Every conversion here goes through degrees as the pivot:
radians and gradians are both converted to degrees first, then
from degrees into whichever unit you asked for. That means only
two real formulas exist —
degrees → radians: deg × (π / 180) and
degrees → gradians: deg × (10 / 9) — and every other
pair (like radians → gradians) is just those two composed, which
keeps the math both correct and easy to audit.
Worked example
180° converts to π radians (≈ 3.14159265 rad) and 200 gradians — half a turn is exactly half of both a full 2π-radian circle and a full 400-gradian circle. A quarter turn, 90°, is π/2 radians (≈ 1.5708 rad) and exactly 100 gradians — the round number that gradians were designed to produce for right angles. And a full 360° turn is 2π radians (≈ 6.28319 rad) and 400 gradians, by definition.
Common angles at a glance
| Degrees | Radians | Gradians |
|---|---|---|
| 0° | 0 rad | 0 grad |
| 15° | 0.2618 rad | 16.6667 grad |
| 30° | 0.5236 rad | 33.3333 grad |
| 45° | 0.7854 rad | 50 grad |
| 60° | 1.0472 rad | 66.6667 grad |
| 90° | 1.5708 rad | 100 grad |
| 120° | 2.0944 rad | 133.3333 grad |
| 135° | 2.35619 rad | 150 grad |
| 150° | 2.61799 rad | 166.6667 grad |
| 180° | 3.14159 rad | 200 grad |
| 270° | 4.71239 rad | 300 grad |
| 360° | 6.28319 rad | 400 grad |
Frequently asked questions
What's a gradian, and why would I ever need one?
A gradian (also called a "gon") divides a full circle into 400 equal parts instead of 360, so a right angle is a clean 100 gradians instead of 90° — that's the entire point of the unit: rounder numbers for right-angle-heavy work. It's used in surveying and some European civil-engineering contexts, where a lot of angles cluster around 90° and 100 is simply easier to work with than 90. You'll rarely meet it outside those fields, but when you do, this tool has it covered as a first-class unit, not an afterthought.
How is a radian actually defined?
One radian is the angle where the arc length around a circle equals the circle's radius — so a full 360° turn, whose total arc length is the circumference 2πr, works out to exactly 2π radians. That's why π keeps showing up: 180° = π radians, 90° = π/2 radians, and so on. Radians are the unit calculus and most programming math libraries use internally (JavaScript's Math.sin, Math.cos, etc. all expect radians), which is the main reason to convert into them.
Why do the results show so many decimal places?
Because radians almost never land on a round decimal — 90° is exactly π/2, but π is irrational, so 1.5707963267948966… never terminates. This tool rounds the displayed value to 6 decimal places, which is precise enough for virtually any practical use (engineering, design, homework) without printing an unreadable wall of digits. The underlying JavaScript math itself uses standard double-precision floating point, the same as any calculator or spreadsheet.
Can I convert negative angles or angles over 360°?
Yes — unlike some inputs on this site, angles aren't restricted to a fixed range here, because negative values and values past a full turn are mathematically normal. A negative angle typically denotes the opposite rotation direction (clockwise vs. counter-clockwise), and anything over 360° (or over 400 gradians, or over 2π radians) just means more than one full turn. Enter whatever number you have — the same degrees/radians/gradians formulas apply regardless of size or sign.