How this instrument works
Degrees split a full rotation into 360 equal slices, an old convention with no deep mathematical necessity behind it. Radians measure an angle instead by the length of arc it traces on a circle of radius 1 — a full rotation there works out to 2π, not 360. Multiplying a degree figure by π over 180 rescales it from the first system into the second, without altering the angle being described in any way.
This site's Triangle Degree calculator runs the identical conversion the opposite way, taking a radian measurement and returning degrees instead; between the two pages, the conversion is covered start to finish in both directions. Reach for this page whenever a protractor, a spec sheet, or a compass bearing hands you degrees but the next calculation — a series expansion, a physics formula, plenty of programming libraries — expects radians instead.
Certain degree values convert to remarkably tidy radian figures because they line up with simple fractions of a full turn: a quarter turn, 90°, lands on π⁄2; a half turn, 180°, lands on π itself; and a full turn, 360°, lands on 2π. Values in between generally convert to messier, non-terminating decimals, which is exactly why a calculator earns its keep here rather than mental arithmetic.
- Enter the angle, measured in degrees, into the Angle in degrees field.
- Read Angle in radians for the identical angle, rescaled onto the radian system.
- Try 90, 180, and 360 to see the tidy landmarks π⁄2, π, and 2π appear directly.
Worked example — 90°, 180°, and 360°
The golden case: 90° times π⁄180 gives π⁄2, roughly 1.570796 radians — a quarter turn, and a right angle in either unit. Double the input to 180° and the result doubles too, landing on exactly π, roughly 3.141593 radians, a straight angle stretched across half a full rotation.
Carry the same input on to 360°, a complete turn back to the start, and the radian figure lands on exactly 2π, roughly 6.283185. Each of these three results is a clean multiple of π specifically because 90°, 180°, and 360° are themselves clean fractions — a quarter, a half, and the whole — of one full rotation.
Questions
What is the formula for converting degrees to radians?
Multiply the degree figure by π divided by 180. Ninety degrees, for example, becomes 90 times π⁄180, which equals π⁄2, near 1.5708 radians, since 90° is a quarter of a full 360° turn.
Why does 180° convert to exactly π?
Because 180° is precisely half of a full 360° rotation, and a full rotation in radians measures exactly 2π — half of that figure is π itself, about 3.14159265 radians, with no rounding involved beyond the digits of π.
How is this different from the Triangle Degree calculator on this site?
That page converts the opposite direction, from radians into degrees; this page starts from degrees and returns the radian equivalent. Together the two pages cover the identical conversion both ways, so whichever unit you're handed, one of them gets you to the other.
Why use radians instead of just sticking with degrees?
Because calculus, physics formulas, and most programming libraries are built around radians natively — derivatives and series expansions of sine and cosine only take their simplest form when the angle is measured in radians, so converting early avoids awkward extra constants later on.
Does 360° always convert to 2π radians?
Yes, exactly — 360° is a complete rotation by definition, and a complete rotation measures out to 2π radians on the arc-length system, roughly 6.283185. Any angle larger than 360° simply keeps adding further multiples of 2π.
Is the conversion exact, or does it round?
The multiplication itself, degrees times π over 180, is exact; any rounding you see comes only from π being an irrational number with infinitely many digits, truncated here to double-precision accuracy — more than enough for any practical measurement.