Introduction to Radians

Recall that dividing a circle into 360 parts creates the degree measurement. This is an arbitrary measurement, and we may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.

The circumference of a circle is 

$C = 2 \pi r$

If we divide both sides of this equation by $r$, we create the ratio of the circumference, which is always $2\pi$ to the radius, regardless of the length of the radius. So the circumference of any circle is $2\pi \approx 6.28$ times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in the diagram below.

The circumference of a circle compared to the radius

The circumference of a circle is a little more than 6 times the length of the radius.

This brings us to our new angle measure. The radian is the standard unit used to measure angles in mathematics. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. 

$ $

One radian

The angle $t$ sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as the length of the radius of the circle. 

Because the total circumference of a circle equals $2\pi$ times the radius, a full circular rotation is $2\pi$ radians.

Radians in a circle

An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to an angle of $2\pi$ radians.

Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius), and the length units cancel. You may sometimes see radians represented by the symbol $rad$.

Comparing Radians to Degrees

Since we now know that the full range of a circle can be represented by either 360 degrees or $2\pi$ radians, we can conclude the following:

$\displaystyle{ \begin{aligned} 2\pi \text{ radians} &= 360^{\circ} \\ 1\text{ radian} &= \frac{360^{\circ}}{2\pi} \\ 1\text{ radian} &= \frac{180^{\circ}}{\pi} \end{aligned}}$

As stated, one radian is equal to $\displaystyle{ \frac{180^{\circ}}{\pi} }$ degrees, or just under 57.3 degrees ($57.3^{\circ}$). Thus, to convert from radians to degrees, we can multiply by $\displaystyle{ \frac{180^\circ}{\pi} }$:

$\displaystyle{ \text{angle in degrees} = \text{angle in radians} \cdot \frac{180^\circ}{\pi} }$

A unit circle is a circle with a radius of 1, and it is used to show certain common angles. 

Unit circle

Commonly encountered angles measured in radians and degrees.

Example

Convert an angle measuring $\displaystyle{ \frac{\pi}{9} }$ radians to degrees.

Substitute the angle in radians into the above formula:

$\displaystyle{ \begin{aligned} \text{angle in degrees} &= \text{angle in radians} \cdot \frac{180^\circ}{\pi} \\ \text{angle in degrees} &= \frac{\pi}{9} \cdot \frac{180^\circ}{\pi} \\ &=\frac{180^{\circ}}{9} \\ &= 20^{\circ} \end{aligned} }$

Thus we have $\displaystyle{ \frac{\pi}{9} \text{ radians} = 20^{\circ} }$.

Measuring an Angle in Radians

An arc length $s$ is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius. 

This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length $s$ to the radius $r$. 

$\displaystyle{ \begin{aligned} s &= r \theta \\ \theta &= \frac{s}{r} \end{aligned} }$

Measuring radians

(a) In an angle of 1 radian; the arc lengths equals the radius $r$. (b) An angle of 2 radians has an arc length $s=2r$. (c) A full revolution is $2\pi$, or about 6.28 radians. 

Example

What is the measure of a given angle in radians if its arc length is $4 \pi$, and the radius has length $$12? 

Substitute the values $s = 4\pi$ and $r = 12$ into the angle formula:

$\displaystyle{ \begin{aligned} \theta &= \frac{s}{r} \\ & = \frac{4\pi}{12} \\ &= \frac{\pi}{3} \\ &= \frac{1}{3}\pi \end{aligned} }$

The angle has a measure of $\displaystyle{\frac{1}{3}\pi}$ radians.