| Degrees (deg) | Radians (rad) |
|---|---|
| 1 Degree | 0.0174532925199 rad |
| 2 Degrees | 0.0349065850399 rad |
| 3 Degrees | 0.0523598775598 rad |
| 4 Degrees | 0.0698131700798 rad |
| 5 Degrees | 0.0872664625997 rad |
| 10 Degrees | 0.174532925199 rad |
| 20 Degrees | 0.349065850399 rad |
| 25 Degrees | 0.436332312999 rad |
| 50 Degrees | 0.872664625997 rad |
| 100 Degrees | 1.74532925199 rad |
| Reference | Degrees (deg) | Radians (rad) |
|---|---|---|
| A right angle | 90 deg | 1.5708 rad |
| A half turn | 180 deg | 3.14159 rad |
| A full turn | 360 deg | 6.28319 rad |
| Earth's axial tilt | 23.44 deg | 0.409105 rad |
The degree is a unit of plane angle equal to one three-hundred-and-sixtieth of a full turn. Its symbol is a small raised circle, and the abbreviation deg is used where that character is unavailable. It is not an SI unit but is accepted for use with the SI, and it remains the ordinary way of stating an angle outside pure mathematics.
The choice of 360 is Babylonian. Their sexagesimal arithmetic worked in sixties, and 360 is close to the number of days in a year, so the sun appears to advance roughly one degree along the ecliptic each day. The number is also unusually convenient: it divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180, so most common fractions of a circle land on whole degrees.
Subdivision follows the same base. A degree splits into 60 minutes of arc and each minute into 60 seconds of arc, which is why latitude and longitude are written in degrees, minutes and seconds. Decimal degrees have largely replaced that notation in software, since a single number is easier to compute with, but navigation charts and astronomical catalogues keep the older form.
Everything from surveying and construction to camera lenses, screw threads and compass bearings is specified in degrees. A right angle is 90, a straight line 180, a full rotation 360, and those three facts carry most everyday geometry without any conversion at all.
The metric system tried to replace it and failed. Revolutionary France introduced the gradian, one hundredth of a right angle, so that a quarter turn was 100 units and a full turn 400. It survives in a few surveying instruments in continental Europe and in the gradian key still found on pocket calculators, but nothing else adopted it. Modern practice has instead moved away from minutes and seconds while keeping the degree itself: satellite navigation, mapping software and geographic databases store latitude and longitude as decimal degrees, so 51 degrees 30 minutes becomes 51.5. The two notations describe the same angle, and confusing one for the other misplaces a position by tens of kilometres.
One degree equals about 0.0174533 radians, 60 minutes of arc, or 3600 seconds of arc.
The radian is the SI unit of plane angle. Its symbol is rad. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius, which makes a full turn equal to two pi radians, or about 6.28319.
Because it is defined as a ratio of two lengths, the radian is dimensionless. That is not a technicality but the whole point. It is the reason the derivative of the sine function is the cosine function, with no stray constant attached, and the reason the small-angle approximation says that the sine of a small angle is very nearly the angle itself. Every one of those results acquires an ugly factor of pi over 180 if degrees are used instead.
The name was coined by James Thomson, brother of Lord Kelvin, in an examination question at Queen's College Belfast in 1873, though the concept had been in use without a name for a century. Roger Cotes had described the advantages of measuring angles in this way in work published in 1722.
Every programming language and mathematical library expects radians. The trigonometric functions in C, Python, JavaScript and their descendants all take radian arguments, which is a routine source of bugs when a value arrives from a user interface in degrees. Angular velocity in physics and engineering is expressed in radians per second, and rotational quantities such as torque and angular momentum follow the same convention.
Rotation is measured in radians per second for the same reason. Angular velocity in radians per second multiplied by a radius in metres gives a speed in metres per second directly, with no conversion factor, so a wheel turning at 100 radians per second on a 0.3 metre radius has a rim speed of 30 metres per second. The same coherence appears in the small-angle approximation, which says that for a small angle expressed in radians the sine, the tangent and the angle itself are nearly equal. That approximation underlies the pendulum formula, the analysis of lenses and half the derivations in an engineering textbook, and it holds only when the angle is in radians.
One radian equals about 57.29578 degrees, and a full circle equals two pi radians, roughly 6.28319.