Degrees vs Radians: Converting Angle Units
Degrees are what everyone thinks in and radians are what the mathematics actually uses. Both measure the same thing and the conversion is one multiplication, but understanding why radians exist makes the whole subject less arbitrary. The Angle Converter handles degrees, radians and gradians.
Why 360 degrees
There is no mathematical reason. It comes from Babylonian astronomy and a base-60 counting system, helped by the fact that 360 divides neatly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more - which makes halves, thirds and quarters of a circle whole numbers. Convenient, and entirely a convention.
Why radians are not arbitrary
A radian is the angle you get when the arc along the circle is the same length as the radius. That definition uses nothing but the circle itself, which is why it falls out of the geometry rather than being chosen.
Go all the way round and the arc is the circumference, which is 2πr - so a full circle is 2π radians, about 6.283. A half turn is π, a right angle is π/2.
The payoff is that formulas get simpler. Arc length is just the radius times the angle in radians. The rate of change of the sine function is the cosine - which is only true in radians; in degrees you pick up an awkward factor of π over 180 everywhere. That is the real reason mathematics uses radians, not tradition.
The conversions
Degrees to radians: multiply by π and divide by 180. Radians to degrees: multiply by 180 and divide by π.
Worth knowing by sight: 30 degrees is π/6, 45 is π/4, 60 is π/3, 90 is π/2, 180 is π, 360 is 2π. One radian is about 57.3 degrees, which is a slightly odd number and the reason nobody uses radians conversationally.
Gradians, the metric angle that did not catch on
A gradian divides a right angle into 100, so a full circle is 400. It was part of the French metric push and it makes a certain sense - a quarter turn being exactly 100 is tidy.
It survives in some European surveying and in the "grad" mode on scientific calculators, which is mostly encountered by accident when a trigonometry answer comes out wrong and nobody can see why. If your sine of 30 is not 0.5, check the mode. The Scientific Calculator makes the current mode visible for exactly that reason.
Common questions
Why does my calculator give the wrong trig answer?
Almost always the angle mode. Degrees, radians and gradians give three different answers for the same input.
Which should I use in code?
Radians. Nearly every programming language's trigonometric functions expect them, and feeding degrees in is a classic bug.
Are minutes and seconds of arc still used?
Yes - in navigation and astronomy. A degree is 60 arcminutes and an arcminute is 60 arcseconds, which is also why a nautical mile is one arcminute of latitude.
Got an angle to convert?