# What is Radian Measure?

##### 1 Answer

Imagine a circle and a central angle in it. If the length of an arc that this angle cuts off the circle equals to its radius, then, by definition, this angle's measure is *1 radian*. If an angle is twice as big, the arc it cuts off the circle will be twice as long and the measure of this angle will be *2 radians*. So, the ratio between an arc and a radius is a measure of a central angle in *radians*.

For this definition of the angle's measure in *radians* to be logically correct, it must be independent of a circle.

Indeed, if we increase the radius while leaving the central angle the same, the bigger arc that our angle cuts from a bigger circle will still be in the same proportion to a bigger radius because of *similarity*, and our measure of an angle will be the same and independent of a circle.

Since the length of a circumference of a circle equals to its radius multiplied by *radians*.

From this we can derive other equivalencies between *degrees* and *radians*: