Radians are an alternate unit to measure angles. Where there are 360 degrees in a full circle, there are $$2\pi$$ radians. The unit for radians is "rad" or a superscript "c" for "circular measure". The latter of these can however easily get confused with the symbol for degrees.

## How big is a radian?

When the arc length between two points on the circumference is equal to the radius of the circle, the angle between those points is 1rad. This means that for a unit circle (radius of 1), an angle's measurement in radians is numerically equal to the length of a corresponding arc.

The circumference of a circle is $$2\pi r$$ where $$r$$ is the radius. This means you can fit $$2\pi$$; radius lengths into the circumference; and so there are $$2\pi$$ radians in a full circle. As can be seen from this diagram, we can fit a little over $$6$$ (or $$2\pi$$) radian-wide sectors into the circle.

To convert from one unit to the other we need a "conversion factor"; this is just a number we multiply by to get from one to the other. It's like saying "What do I need to do to get from $$2\pi$$ to $$360$$?". Let's call this conversion factor $$x$$, such that $$2\pi x=360$$. We can easily see from rearranging that $x=\frac{360}{2\pi}=\frac{180}{\pi}$ So to convert from radians to degrees we multiply by $$\frac{180}{\pi}$$.
This is just the reverse of the above conversion. There's some $$y$$ such that $$360y=2\pi$$, solving gives $y=\frac{2\pi}{360}=\frac{\pi}{180}$ So to convert from degrees to radians, we multiply by $$\frac{\pi}{180}$$.