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Angle Converter
Convert between degrees, radians, gradians, and arc-minutes. Essential for trigonometry, navigation, engineering, and physics. Learn why radians are the mathematical standard and how to convert between all angle units.
Conversion Formula
Radians = Degrees x pi / 180 | Degrees = Radians x 180 / pi | Gradians = Degrees x 10/9
Angle Unit Conversion Guide
Angles are measured in three main units: degrees (everyday use), radians (mathematics and physics), and gradians (surveying). The degree system (360 per circle) dates to ancient Babylon. Radians are the SI standard used in all advanced mathematics.
Degrees — The Everyday Angle
A full circle = 360 degrees. Right angle = 90 degrees. Straight angle = 180 degrees. Each degree is divided into 60 arc-minutes and 3,600 arc-seconds — used in GPS coordinates and astronomy.
Radians — The Mathematical Standard
1 radian is the angle subtended by an arc equal to the radius. A full circle = 2*pi radians. pi radians = 180 degrees. Calculus and physics formulas require radians — the derivative of sin(x) is cos(x) ONLY when x is in radians.
Gradians
A gradian (gon) divides a right angle into 100 units, making a full circle 400 gradians. Used in some European surveying systems. 1 degree = 1.1111 gradians.
Angle
Degrees
Radians
Gradians
Full circle
360
2*pi (6.2832)
400
Straight angle
180
pi (3.1416)
200
Right angle
90
pi/2 (1.5708)
100
1 degree
1
0.017453
1.1111
Frequently Asked Questions
Multiply degrees by pi/180 (approximately 0.017453). To convert radians to degrees, multiply by 180/pi (approximately 57.2958). Example: 90 degrees = 90 x pi/180 = pi/2 = 1.5708 radians.
pi radians = 180 degrees exactly. 2*pi radians = 360 degrees. pi/2 radians = 90 degrees. pi/4 = 45 degrees. pi/6 = 30 degrees. pi/3 = 60 degrees. These are the key angle relationships in trigonometry.
Radians simplify calculus formulas. The derivative of sin(x) is cos(x) only when x is in radians. All Taylor series for trig functions assume radians. Physics and engineering equations are naturally simpler in radians.
1 radian = 180/pi = approximately 57.2958 degrees. A full circle has 2*pi = 6.2832 radians. One radian is the angle where the arc length equals the circle radius — about 57.3 degrees or roughly 1/6 of a circle.
A gradian (gon) divides a right angle into 100 units, making a full circle 400 gradians. This base-10 system simplifies some surveying calculations. 1 gradian = 0.9 degrees = pi/200 radians.
1 degree = 60 arc-minutes = 3,600 arc-seconds. Arc-seconds are used in astronomy and GPS precision. The Hubble Space Telescope resolves detail at about 0.1 arc-seconds — about 1/36,000 of a degree.