Angle Converter

A full turn is 360 degrees or 2π radians, which gives the whole conversion:

• Degrees to radians: multiply by π/180
• Radians to degrees: multiply by 180/π
• 1 radian = 57.29578°; 1° = 0.0174533 rad

The familiar landmarks are worth holding in radians: 90° is π/2, 180° is π, 60° is π/3, 45° is π/4, 30° is π/6.

The 360 is arbitrary, inherited from Babylonian base-60 arithmetic and convenient because it divides so many ways. A radian is not arbitrary at all: it is the angle where the arc equals the radius.

Because a radian is defined as a ratio of two lengths, it is dimensionless, and that is what makes the standard results work.

The derivative of sin x is cos x only when x is in radians. In degrees the derivative picks up a factor of π/180. The small-angle approximation sin x ≈ x is likewise a radian statement: sin(0.01 rad) is 0.0099998, while sin(0.01°) is 0.000175.

This is also the most common cause of a wrong trigonometry answer. sin(30) is 0.5 in degree mode and −0.988 in radian mode. If the answer is inexplicable, check the calculator's mode before checking the working.

Three smaller traditions persist because each suits its job:

• Gradian (gon): 400 to a full turn, so a right angle is a round 100. A metric tidying-up of the degree that survives in surveying.
• Arcminute: one sixtieth of a degree
• Arcsecond: one sixtieth of an arcminute, so 1/3600 of a degree

The sixtieths come from the same Babylonian arithmetic as the 360, and they remain standard in astronomy and navigation, where positions are quoted to arcseconds.

There is a neat consequence at sea: one arcminute of latitude is one nautical mile, which is why that unit is 1,852 metres rather than a round number. The nautical mile was defined from the Earth, not the other way round.