Introduction: What is a radian? A radian is a unit of measuring angles, yes, but where does it come from exactly?

The answer to this is born out of a thought experiment: what is the angle formed when a central angle of a circle intercepts an arc whose length is also exactly equal to the radius:

The name itself is an acknowledgement of the process by which the angles is created: radian ~ radius. 

Contrary to the units for angles you may use more regularly, the degree, the radian is the official SI unit of angle measurement. The origins of the degree as a unit of measurement are not well understood, although there are many theories relating to astronomy and the near alignment of 360~365 days in a year. It also could be the result of a base 20 counting system or subdividing the circle into equilateral triangles, but the true origins are unknown.

While you may already know that the full circle encompasses 360 degrees, how many radians does the circle contain? In this case, we can look to the circumference ask the question: how many times does the radius fit into the circumference of the circle? This will also answer the question of how many radians a circle encompasses since each radian corresponds to exactly to 1 radius along the circumference of the circle. Of course, the circumference of the circle is known to be \(2\pi r\), which quickly answers the question: there are exactly \(2\pi\) radians in one full circle. 

This leads to the conversion between radians and degrees, since both 360 degrees and \(2\pi\) radians exactly cover the same circle:

\[\solve{2\pi \text{ radians} &=& 360^\circ\\\pi \text{ radians}&=& 180^{\circ}}\]

By using this, we can convert from degrees to circles by multiplying any degree measure by: \[\dfrac{\pi}{{180}}\] Similarly, we can convert any radian measure to degrees by multiplying by:\[\dfrac{{180}}{\pi}\]