How this calculator works
An angle measures rotation. The radian is the SI unit and is defined so that one full turn is 2 pi radians, which makes angular formulas in physics and calculus clean. A degree is one 360th of a turn, a gradian is one 400th of a turn, and a turn is the full rotation. Minutes and seconds are subdivisions of a degree: one degree has 60 minutes and one minute has 60 seconds. A milliradian is one thousandth of a radian and is common in optics and artillery. Negative values describe rotation in the opposite direction.
Formula
radians = value x F(unit), where F is: deg pi/180, rad 1, gon pi/200, turn 2 pi, arcmin pi/10,800, arcsec pi/648,000 and mrad 0.001. For degrees, minutes and seconds, radians = sign(degrees) x (|degrees| + minutes/60 + seconds/3,600) x pi/180. Degrees = radians x 180/pi. Pi radians = radians/pi.
Worked example
45 degrees is 0.785398 rad, 50 gradians, 0.125 turns, 0.25 pi rad, 2,700 arcminutes and 162,000 arcseconds. 1 radian is 57.2958 degrees, 63.662 gradians and 3,437.75 arcminutes. 45 degrees 30 minutes 0 seconds is 45.5 degrees.
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Frequently asked questions
Why do mathematicians use radians instead of degrees?
In radians, arc length is radius times angle and the derivative of sine is cosine without a scale factor. Degrees and gradians are convenient for surveying and everyday angles, but they carry an arbitrary factor into every formula.
How many gradians are in a right angle?
One right angle is 100 gradians, because a full turn is 400 gradians. The gradian was introduced with the metric system and is still used in some surveying instruments.
How do I read 45 degrees 30 minutes 0 seconds?
It means 45 whole degrees plus 30 sixtieths of a degree, which is 45.5 degrees. The seconds part adds sixtieths of a minute, so 45 degrees 30 minutes 30 seconds is 45.508333 degrees.