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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.
Here are some quick reference conversions from radian (rad) to degree (deg):
| radians | degrees |
|---|---|
| 0.000001 rad | deg |
| 0.001 rad | deg |
| 0.1 rad | deg |
| 1 rad | deg |
| 2 rad | deg |
| 3 rad | deg |
| 4 rad | deg |
| 5 rad | deg |
| 6 rad | deg |
| 7 rad | deg |
| 8 rad | deg |
| 9 rad | deg |
| 10 rad | deg |
| 20 rad | deg |
| 30 rad | deg |
| 40 rad | deg |
| 50 rad | deg |
| 100 rad | deg |
| 1000 rad | deg |
| 10000 rad | deg |
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.