Degrees, radians and gradians
A full circle can be cut up in several ways. The degree splits it into 360 parts, a choice inherited from Babylonian astronomy that survives because 360 divides evenly by so many numbers. The radian is the natural mathematical unit: one radian is the angle that subtends an arc equal to the radius, so a full circle is exactly 2 pi radians and one radian is 57.29578 degrees. Every trigonometric formula in calculus assumes radians, and so does the Math library in JavaScript, Python and most spreadsheets when you call sin or cos. To convert, multiply degrees by pi/180, or multiply radians by 180/pi.
The gradian, also called the gon, divides the circle into 400 parts so that a right angle is a clean 100. It appears in European surveying and on scientific calculators with a GRA mode. Finally, angles can be counted in turns or revolutions, which is how rotation is described mechanically: a quarter turn is 90 degrees, and 3,000 rpm is 3,000 turns per minute.
Minutes, seconds and coordinates
Degrees subdivide the same way as an hour: one degree is 60 arcminutes and one arcminute is 60 arcseconds, so a degree holds 3,600 arcseconds. Latitude and longitude are still written this way on charts and older maps. To convert degrees-minutes-seconds to decimal degrees, divide the minutes by 60, the seconds by 3,600, and add them to the whole degrees, keeping the sign of the hemisphere. For example 40 degrees 26 minutes 46.3 seconds is 40 + 26/60 + 46.3/3600 = 40.446194 degrees.
Precision matters here. One degree of latitude is about 69 miles, one arcminute is about 1.15 miles (one nautical mile, by definition), and one arcsecond is about 101 feet. Six decimal places in decimal degrees locates a point to roughly four inches, which is more than a phone GPS can deliver. Astronomers keep going: the apparent diameter of the full Moon is about 31 arcminutes, and a good telescope resolves detail measured in single arcseconds.
Worked examples and practical uses
Math: 45 degrees is pi/4 = 0.785398 rad; 90 degrees is 1.570796 rad; 180 degrees is 3.141593 rad; and 1 rad is 57 degrees 17 minutes 45 seconds. Construction: roof pitch is quoted as rise over run, so a 6:12 pitch is an angle of 26.57 degrees, which the converter also reports as a 50 percent slope. Highway grades are given as percent slope, where 6 percent is 3.43 degrees.
Mechanics and optics: a bolt tightened an extra 90 degrees has turned a quarter turn; a camera with a 60 degree horizontal field of view covers one sixth of the horizon; a laser specified at 1 milliradian of divergence spreads about one meter over a kilometer, which is 3.44 arcminutes. When you feed angles into a spreadsheet, wrap degrees in the RADIANS function first, since SIN and COS expect radians and silently return nonsense otherwise.