One complete circle.
Circles to Radians Converter — circle to rad
Convert Circle (circle) to Radian (rad) using the exact conversion factor (1 circle = 6.283185307 radian). See the formula, worked examples, conversion table, history, and common uses.
Circles to Radians converter
Converter
Angle Converter
Convert radians, degrees, gradians, turns, arcminutes, arcseconds, and milliradians.
Context values
Some unit families need one extra value for a physically accurate result.
The radian is the SI angle unit.
Result = input x 6.283185307 / 1.
Angle uses radians as the base unit.
Reserved above the conversion cards and below the converter.
About Converting Circles to Radians
Circle and Radian are the two standard ways to measure angles — degrees for everyday and navigational use, radians for calculus and physics — so converting between them comes up constantly in math, engineering, and programming. Specifically: One complete circle, and the radian is the SI angle unit.
Formula
radians = circles × 6.28318530718
This factor comes from each unit's defined relationship to the category's base unit: 1 circle equals 6.28318530718 base units, and 1 radian equals 1 base unit, so dividing one by the other gives the direct circle-to-radian factor of 6.28318530718.
Simple example
1 circle × 6.283185307 = 6.283185307 rad
1 circle = 6.283185307 radians.
Real-world example
90 circle × 6.283185307 = 565.4866776 rad
90 circles = 565.4866776 radians.
Conversion table
| Circle (circle) | Radian (rad) |
|---|---|
| 0 circle | 0 rad |
| 30 circle | 188.4955592 rad |
| 45 circle | 282.7433388 rad |
| 90 circle | 565.4866776 rad |
| 180 circle | 1,130.973355 rad |
| 360 circle | 2,261.946711 rad |
Reverse conversion: Radian to Circle
6.283185307 rad × 0.1591549431 = 1 circle
6.283185307 radians = 1 circles.
circles = radians × 0.159154943092
For a page dedicated to this direction, see Radian to Circle.
Understanding the Circle (circle)
One complete circle.
Understanding the Radian
Radian (rad) measures angle. A radian is the angle subtended when the arc length equals the radius; a full circle is 2π radians. It is classified as a SI derived unit.
Where it's used: Standard angle unit in mathematics, physics, and engineering worldwide.
Uses of the Radian today
- Calculus and trigonometry (derivatives of trig functions are simplest in radians)
- Physics and engineering (angular velocity, rotational mechanics)
- Computer graphics and robotics
History of the Radian
The radian was developed in the 1870s, with the term itself coined by James Thomson (brother of Lord Kelvin) around 1873, formalizing a ratio-based angle measure that mathematicians had used implicitly for calculus since Newton and Leibniz.
It was adopted as the SI derived unit of angle because it makes many physics and calculus formulas simpler than using degrees.
Historical origin. Radian was not the invention of a single named person; it developed through the process described above.
Radian — sources
- BIPM SI Brochure — Coherent derived units — Defines the radian as the SI unit of plane angle.
Reserved between the cards and the FAQ so the page stays balanced.
Frequently asked questions
How many radians are in 1 circle?
1 circle equals 6.283185307 radians, using the exact defined conversion factor rather than an estimate.
How do I convert circle to radian?
Multiply the circle value by 6.283185307. The converter above does this instantly to whatever precision you set.
How do I convert radian back to circle?
Use the reverse factor: 1 radian equals 0.1591549431 circles. You can also use the swap control in the converter above to flip the direction instantly.
What is a circle?
Circle (circle) is a unit of angle.
What is a radian?
Radian (rad) measures angle. A radian is the angle subtended when the arc length equals the radius; a full circle is 2π radians. It is classified as a SI derived unit.
Is the circle to radian conversion exact?
Yes. Both circle and radian are defined by fixed standards rather than physical artifacts, so the factor of 6.283185307 used above is exact to as many digits as you choose to display.