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To convert a specific radian measure into degrees, it helps to understand the relationship between these two units. A complete revolution of a circle is 2π radians, which is equivalent to 360 degrees. Simplifying this relationship, π radians is exactly equal to 180 degrees.
Because the radian is a derived unit in the International System of Units (SI), converting it to the more familiar unit of angular measure, the degree, requires a simple calculation.
To calculate the value in degrees π is included in the conversion ratio. The formula is:
Let's say you want to convert an angle of π/6 radians.
You can perform this calculation manually for any angle, or simply use our radians to degrees calculator at the top of this page for instant, precise results.
How many radians are in one degree? One degree (deg) contains 0.017453292519943294444 radians (rad) — the inverse of the factor above. Multiplying by 57.295779513082325226 takes you from radians to degrees; multiplying by 0.017453292519943294444 brings you back.
Put in words: one radian equals 57.295779513082325226 degrees, so the radian is the larger unit of this pair. Converting between them never changes the amount of angle being measured — only the size of the unit you count it in.
A radian (rad) is the standard unit used to measure angles, and it's part of the International System of Units (SI).
One radian is equal to roughly 57.3 degrees (deg), or more precisely, 180/π degrees.
The radian is the official SI unit for angles, making it the preferred choice in science and engineering.
It is especially common in fields like physics, calculus, and computer graphics. The main advantages of using radians are:
A full circle has 360 degrees (360°), which is equal to 2π radians.
This key relationship gives us the precise formulas for converting between radians and degrees:
Degrees = Radians × (180/π)Radians = Degrees × (π/180)Here are some common angle conversions:
| Degrees | Radians (exact) | Radians (approx.) |
|---|---|---|
| 30° | π/6 rad | 0.524 rad |
| 45° | π/4 rad | 0.785 rad |
| 60° | π/3 rad | 1.047 rad |
| 90° | π/2 rad | 1.571 rad |
| 180° | π rad | 3.142 rad |
| 360° | 2π rad | 6.283 rad |
Geometrically, the definition of a radian is based on a circle's own properties.
Imagine an arc along the edge of a circle that has the same length as the circle's radius. The angle created at the center of the circle by this specific arc is exactly one radian.
Because a radian is a ratio of two lengths (arc length divided by radius), it is technically a dimensionless unit. This unique property is why it integrates so smoothly into advanced math and physics formulas.
In geometry and everyday life, we measure angles using degrees.
A degree (represented by the universal symbol °) is the basic unit for measuring rotation. It helps us describe the amount of turn between two lines that meet at a point.
For instance, a perfect corner, like the edge of a book, is a 90° right angle, and one complete, full-circle rotation measures 360°.
The standard of a circle containing 360 degrees (360°) originated with the ancient Babylonians.
They used a base-60 (sexagesimal) number system, and 360 was a perfect number for them because it is highly divisible. Its large number of factors made it incredibly easy to perform fractional calculations in fields like astronomy and geometry.
For fields that require extreme precision, like astronomy, cartography (map-making), and GPS navigation, a single degree is broken down into even smaller units:
This system ensures that even the tiniest angles can be measured with high accuracy.
There is no physical reason a full turn must be 360 of anything. The number is an inheritance from Babylonian astronomy, and it survived for two reasons that still matter.
The first is the calendar. Babylonian astronomers worked with a year of roughly 360 days, so the Sun appeared to advance about one degree along the ecliptic each day — a convenient bookkeeping unit for tracking the sky.
The second is arithmetic, and it is the reason the convention outlived the astronomy. 360 has 24 divisors. It splits evenly into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths, twelfths, fifteenths, eighteenths, twentieths, twenty-fourths and more. Every common fraction of a turn lands on a whole number of degrees: a right angle is 90°, a third of a turn is 120°, an eighth is 45°. A base-10 alternative such as the gradian, with 400 to a circle, is better for decimal arithmetic but produces fractions for thirds and sixths — which is roughly why it never displaced the degree outside surveying.
Degrees are for people; radians are for mathematics and for machines.
The distinction is not stylistic. In calculus the derivative of sin(x) is cos(x) only when x is in radians; in degrees an awkward factor of π/180 appears and never leaves. Because of this, essentially every programming language's trigonometric library — JavaScript, Python, C, Java — takes radians, not degrees.
That mismatch is a reliable source of bugs. A value that looks plausible but is wrong by a factor of about 57.3 usually means degrees were passed where radians were expected. The conversions worth memorising are:
Scientific calculators expose the same trap through their DEG / RAD / GRAD mode switch, where an unnoticed mode is the classic cause of an answer that is confidently and completely wrong.
Here are some quick reference conversions from radian (rad) to degree (deg):
| radians | degrees |
|---|---|
| 0.000001 rad | 0.000057295779513082325226 deg |
| 0.001 rad | 0.057295779513082325226 deg |
| 0.1 rad | 5.7295779513082325226 deg |
| 1 rad | 57.295779513082325226 deg |
| 2 rad | 114.59155902616465045 deg |
| 3 rad | 171.88733853924697568 deg |
| 4 rad | 229.1831180523293009 deg |
| 5 rad | 286.47889756541162613 deg |
| 6 rad | 343.77467707849395136 deg |
| 7 rad | 401.07045659157627658 deg |
| 8 rad | 458.36623610465860181 deg |
| 9 rad | 515.66201561774092703 deg |
| 10 rad | 572.95779513082325226 deg |
| 20 rad | 1145.9155902616465045 deg |
| 30 rad | 1718.8733853924697568 deg |
| 40 rad | 2291.831180523293009 deg |
| 50 rad | 2864.7889756541162613 deg |
| 100 rad | 5729.5779513082325226 deg |
| 1000 rad | 57295.779513082325226 deg |
| 10000 rad | 572957.79513082325226 deg |
Engineering datasheets across Europe write radian to degree several different ways. The calculation is the same in every one of them.
| Language | Written as | Unit names |
|---|---|---|
| French | rad en deg | radian → degré |
| Spanish | rad a deg | radián → grado |
| Dutch | rad naar deg | radiaal → graad |
| German | rad in deg | Radiant → Grad |
| Polish | rad na deg | radian → stopień |