Radian Measure

A Bit of History

With a bit of research into degrees as a measure of angle one might find that the system you've known and loved since grade school is several thousand years old and based on Babylonian superstition. They had a religion that loved numbers (nerds!) and were fascinated by numbers with lots of factors - they had a sexagesimal number system. The number 60 has lots of factors… And, well, 6 times 60 is 360 which has even more factors! From these folk we all got the idea that there are 360 degrees in a circle.

For other theories check out the most beloved wikipedia.

So the the idea to define angles based on geometry seems like a good one. It occurred to Roger Cotes around 1714. It took another 150 odd years to come up with a name. Math folk can be a bit slow…

A Few Definitions to Get Started

Sector - The portion of a circle enclosed by two radii and an arc.

Arc Length - The distance along the curved line making up the arc. Imagine wrapping a shoe lace around a curve then measuring the length of the lace…


Basic Idea

The angle of a sector is determined by the ratio of the arc length and the radius. That's it. Easy to say, but not so easy to understand.

If the ratio of the arc-length to radius is fixed so is the angle. For example take the two sectors below. Note that for both $\frac{arc-length}{radius}=0.5$ and that the angles are both the same! Don't believe me? Grab a protractor and draw a few for yourself.


The ratio of the arc-length to the radius determines the angle. This gives a new and geometrical measure of the angle. So using the notation that $l=arc-length$ and $r=radius$ we can say:

\begin{align} \frac{l}{r}=\theta \end{align}

When this ratio is 1 then we'd say we have an angle of 1 radian.

\begin{align} \frac{l}{r}=1 => 1 \: radian \end{align}

Converting Between Radian and Degrees

When learning a new system of measure its helpful to be able to convert back to the old system in order to get a feeling for what's what. So lets consider a few simple examples.

A full circle is $360^\circ$. If the radius is r then the circumference (or arc-length) is $2 \pi r$. So that means there are $2 \pi$ radians in a full circle. That is:

\begin{align} \frac{2 \pi r}{r}=2 \pi \end{align}

Thus half a circle is only $\pi$ radians. From this we can say

\begin{align} 180^\circ = \pi \end{align}

This provides us with a conversion factor (which always equal 1). That can be written two ways depending on what is needed.

\begin{align} \frac{180^\circ}{\pi}=1 \end{align}


\begin{align} \frac{\pi}{180^\circ}=1 \end{align}

The first is used to convert from radians to degrees. For example $\frac{\pi}{3}$ equals:

\begin{align} \frac{\pi}{3} \frac{180^\circ}{\pi} = 60^\circ \end{align}

The second is used to convert from degrees to radians. For example $30^\circ$ equals:

\begin{align} 30^\circ \frac{\pi}{180^\circ} = \frac{\pi}{6} \end{align}

Note radians are unit-less measure (length divided by length). Occasionally the "unit" of rad is used to make it extra clear that the number refers to radians.

Table of Equivalence


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