Enter an angle in degrees, radians, gradians, minutes or seconds and the converter shows it in the others.
Most trouble with angles comes from calculators set to the wrong mode – if sin(30) doesn't give 0.5 you are in radians.
Units of Angle
Degrees divide a circle into 360 parts. The radian is the angle whose arc is as long as the radius, so a circle holds 2π of them and one radian is about 57.3°. Radians are the unit that maths and physics formulas expect.
Gradians (gon) divide a circle into 400 parts, so a right angle is 100 gon. Minutes and seconds of arc are 1/60 and 1/3,600 of a degree, and astronomers use them for the tiny angles between stars.
Variations of Milliradians
Milliradians (mrad, also called mils) are one thousandth of a radian, about 0.057°, and are used for small angles in military sights. Strictly a circle holds 6,283 of them, so armies round to a convenient whole number and the variants differ.
The NATO mil rounds the circle to 6,400 parts of 0.05625° each, which makes a quarter turn a clean 1,600 mils.
The Swedish mil, or streck, divides the circle into 6,300 parts of about 0.0571° each, and was used for artillery.
About Degrees
A degree (°) is 1/360 of a full turn, and an angle is measured by how far one line has rotated from another. The 360 is generally credited to the Babylonians, whose base-60 counting made it a convenient choice.
Each degree is split into 60 minutes (′) and each minute into 60 seconds (″), so 1° is 3,600″.
Multiply radians by 180/π to get degrees – 1 radian is about 57.3° – and multiply degrees by π/180 to go back, so 180° is π radians.
About Radians
The radian is the SI unit of angle. A full circle is 2π (about 6.28) radians, against 360 degrees.
A radian is a ratio of two lengths, arc over radius, so it carries no units of its own and drops out of physics formulas – speed = radius × angular velocity works directly when the angle is in radians.
About Gradians
A gradian (also called a gon or grad) is 1/400 of a circle, so a right angle is exactly 100 gradians. It was introduced in France alongside the metric system as a decimal alternative to degrees.
Gradians are mostly found on surveying instruments in continental Europe, which are often marked in gons. One gradian is 0.9 degrees, or π/200 radians.
On a gradian scale a quarter turn is 100 and a half turn is 200, so right angles come out as round numbers. The catch is that 30° and 60° become 33.33 and 66.67 gradians.