📐 Converters

Degrees ↔ Radians (with table)

Convert angles between degrees and radians. Includes a table of common values (0, 30, 45, 60, 90, 180, 270, 360).

Degrees Radians

Tabela de valores comuns:

Degrees	Radianos (exato)	Decimal
0°	0	0
30°	π/6	0.5236
45°	π/4	0.7854
60°	π/3	1.0472
90°	π/2	1.5708
180°	π	3.1416
270°	3π/2	4.7124
360°	2π	6.2832

Degrees and radians, both ways

A degree splits the turn into 360 parts, a number inherited from Babylon for having many divisors. A radian defines the angle from the circle itself: one radian is the angle whose arc equals the radius, which makes a full turn two pi radians. The choice is not arbitrary — in radians, the formulas of calculus come out clean.

Type into either field and the other follows. The references worth carrying: half a turn is pi, a quarter is pi over two, 60 degrees is pi over three and 45 is pi over four. One radian alone is roughly 57.3 degrees, a number that comes up often enough to be worth memorising.

Why the radian is the natural unit becomes clear in calculus: the derivative of sine is cosine only when the angle is in radians. In degrees, a factor of pi over 180 appears in every derivative. It is also why practically every programming library expects radians in its trigonometric functions — passing degrees without converting is among the commonest bugs in geometry code.

Frequently asked questions

Why 360 degrees and not 100?

Babylonian inheritance and convenience: 360 has 24 divisors, which makes halves, thirds, quarters, fifths, sixths and eighths of a turn come out whole. The attempt to decimalise exists — the gradian splits the turn into 400 — but it never caught on outside surveying.

Why do libraries use radians?

Because trigonometric functions are defined in radians mathematically, and implementations follow the definition. Converting at the library boundary would pick an arbitrary convention and surprise anyone doing calculus. Converting is the caller's job, and that is where the bug happens.

What is a gradian?

A unit splitting the turn into 400 parts, so a right angle is exactly 100. It was born alongside the metric system and survives in surveying in some European countries, and as the third option on scientific calculators — where it has caused plenty of wrong answers by being selected unintentionally.

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