We know how to measure the range of an angle using degrees, which can also be divided into the submultiple minutes and seconds.

But there is another way of measuring angles. It can be done by using units called radians.

One radian is the angle obtained when the radius is taken and put around the circle. Let's see an illustration to understand it better:

So, a radian denotes an angle in which its corresponding arc has the same length as its radius. And so, a full angle has $2\pi$ radians, a straight angle has $\pi$ radians, and a straight angle has $\dfrac{\pi}{2}$ radians.

This is deduced from the fact that the total length of a circumference is:

$L=2 \cdot \pi \cdot r$

where $r$ is the radius of such circumference.

Therefore, a full rotation is $2\pi$ times the length of the radius, and considering that a full rotation is $360^\circ$, now we have a way of changing from one measure to another: $2 \cdot \pi$ radians $=360^\circ$ (a whole turn).

The conversion factors that we will use to change from one to another will be:

• to convert from degrees to radians:

$N^\circ=N^\circ \cdot \dfrac{2\pi \ \mbox{radians}}{360^\circ}= \dfrac{N \cdot 2 \pi}{360}$ radians, where $N$ is the number of degrees that we want to express in radians.

• to convert from radians to degrees:

$M$ radians = $M \ \mbox{radians} \cdot \dfrac{360^\circ}{2 \pi \ \mbox{radians}}= \Big(\dfrac{M \cdot 360}{2 \pi} \Big)^\circ$ where $M$ is the number of radians that we want to express in degrees.

Let's write $270^\circ$ in radians:

Taking the degrees conversion factor to radians we have: $$270^\circ \cdot \frac{2\pi \ \mbox{radians}}{360^\circ}= \frac{270 \cdot 2 \pi}{360} \ \mbox{radians}= \frac{3}{2} \pi \ \mbox{radians}$$

When we express the quantities in $radians$, we usually write $\pi$ instead of its value in numbers.

If one is going to put it in number form, rounding $3,1416$ will be fine.

For example:

$$\frac{3}{2} \pi = \frac{3}{2} \cdot 3,1416 = 4,71225 \ \mbox{radians}$$.

Let's write $45^\circ$ in radians: $$45^\circ \cdot \frac{2 \pi \ \mbox{radians}}{360^\circ}=\frac{45 \cdot 2\pi}{360} \ \mbox{radians}= \frac { \pi}{4} \ \mbox{radians}$$ that in numbers would be approximately:$$\frac { \pi}{4}= \frac{3,1416}{4}=0,7853 \ \mbox{radians}$$

Now let's write $3\pi$ radians in degrees:

Like before, we take the conversion factor, but now the one that takes us from radians to degrees, and we obtain:

$$3\pi \ \mbox{radians}= 3\pi\cdot\frac{360^\circ}{2\pi}=540^\circ$$

Let's write $\dfrac {6\pi}{5} \ \mbox{radians}$ in degrees:

$$\dfrac{6}{5}\pi \ \mbox{radians}= \frac{6}{5}\pi \frac{360^\circ}{2\pi}=216^\circ$$