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A radian is a unit for measuring angles, usually written as "rad". When set as a central angle in a circle, a one radian angle will cut an arc with length equal to the radius of the circle. One radian roughly equals 57.2958°. In the image to the right, φ is a 1 rad angle.

A circle's circumference is 2 π times its radius, so an angle that cuts, or subtends, an arc consisting of a circle's entire circumference (360°) is 2 π radians. Because of this relationship between radians and the number π, it is often simplest to measure radians in terms of π. For instance:

φ is a 1 rad angle.

 $2 \pi \;\text{rad}$ $=$ $360^\circ$ $\pi \;\text{rad}$ $=$ $180^\circ$ $\tfrac{\pi}{2} \;\text{rad}$ $=$ $90^\circ$ $\tfrac{\pi}{3} \;\text{rad}$ $=$ $60^\circ$ $\tfrac{\pi}{4} \;\text{rad}$ $=$ $45^\circ$ $\tfrac{\pi}{6} \;\text{rad}$ $=$ $30^\circ$ $0\; \text{rad}$ $=$ $0^\circ$

All of these can be used to convert between degrees and radians, though the one used most often is π rad = 180°. Although radians are most often thought about in terms of the number π, the conversion 1 rad ≈ 57.2958° can also be used to convert between the two.

When converting, the two entities are set as a ratio. The same is done when we multiply the amount of time we've been driving (in hours) by the speed of our car (in miles/ hour) to find the distance we have driven (in miles). The speed is a conversion factor between time and distance—a ratio that relates an amount of time to its corresponding amount of distance—allowing conversion between the two. $\tfrac{180^\circ}{\pi~\text{rad}}$ expresses degrees per radian and is used to convert from radians to degrees, while $\tfrac{\pi~\text{rad}}{180^\circ}$ expresses radians per degree and converts degrees into radians. We then multiply the entity we wish to convert by the appropriate ratio, just as in the time, speed, and distance example.

If you have an arbitrary angle θ° that you want to convert to radians the following method is used:

$\phi^\circ \left (\frac{\pi~\text{rad}}{180^\circ} \right )=\frac{(\phi^\circ)~\pi~\text{rad}}{180^\circ}=\frac{(\phi^{\color{Red}\circ})~\pi~{\color{Blue}\text{rad}}}{180^{\color{Red}\circ}}=\frac{(\phi)~\pi~{\color{Blue}\text{rad}}}{180}=\left (\frac{(\phi)~\pi}{180} \right ) ~{\color{Blue}\text{rad}}$

If you have an arbitrary angle &phi rad that you want to convert to degrees the following method is used:

$\theta~\text{rad} \left (\frac{180^\circ}{\pi~\text{rad}} \right )=\frac{(\theta~\text{rad})~180^\circ}{\pi~\text{rad}}=\frac{(\theta~{\color{Red}\text{rad}})~180^{\color{Blue}\circ}}{\pi~{\color{Red}\text{rad}}}=\frac{(\theta)~180^{\color{Blue}\circ}}{\pi}=\left (\frac{(\theta)~180}{\pi} \right )^{\color{Blue}\circ}$

Notice that in each equation, the units in red end up canceling each other.

Since the circumference of a circle is always 2π times the radius, the measure of an angle in radians is the same regardless of the size of the circle. Indeed, a radian is just a number. It is simply the ratio of the length of an arc subtended by an angle to the length of the radius of the circle that the angle is in. As a result, radians are said to be dimensionless. Polar Coordinates are usually written in radians.