Explaining Radians

What is a radian?

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.

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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.

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Converting radians to degrees

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 \(\frac{180}{\pi}\).

Converting degrees to radians

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 \(\frac{\pi}{180}\).

Interactive Radians Tool

interactive radians

Click here for an interactive tool which will help you get to grips with radians! You can convert between degrees and radians in a simple way, seeing the size of the angle as you change it.