Angle Converter

Multiply degrees by pi/180. Example: 90 degrees = 90 x pi/180 = pi/2 radians ≈ 1.5708. Or use the proportion: degrees/180 = radians/pi. Key angles: 30deg=pi/6, 45deg=pi/4, 60deg=pi/3, 90deg=pi/2, 180deg=pi, 360deg=2pi.

A gradian divides a full circle into 400 gradians (100 per right angle). Used mainly in surveying: a right angle = 100 gradians. Conversion: gradians = degrees x 400/360 = degrees x 10/9. A 90-degree angle = 100 gradians. Gradians make compass bearings easier to calculate.

1 degree = 60 arcminutes (symbol: prime). 1 arcminute = 60 arcseconds (symbol: double-prime). These tiny units measure precise angles in astronomy and navigation. The moon is about 0.5 degrees = 30 arcminutes in diameter. GPS accuracy is expressed in arcseconds.

Radians make calculus simpler: d/dx[sin(x)] = cos(x) only holds when x is in radians. If you use degrees, you get d/dx[sin(x)] = (pi/180)cos(x). Radians also give natural values: arc length = r x theta (in radians). Period of sin: 2pi radians, not 360 degrees.

0 deg = no rotation. 30 deg = pi/6 rad (1/12 circle). 45 deg = pi/4 rad (1/8 circle). 60 deg = pi/3 rad (1/6 circle). 90 deg = pi/2 rad (1/4 circle = right angle). 180 deg = pi rad (straight line). 270 deg = 3pi/2 rad. 360 deg = 2pi rad (full circle).