#### What is the deal with radians anyway?

Most of you are used to thinking of a circle in terms of degrees: 360° is the whole circle. 180° is half the circle etc... Well, radian measure is just a different way of talking about the circle. Radian measure is just different unit of measure.

Just as we can measure a football field in yards or feet--we can measure a circle in degrees (like the good old days) or in radians (welcome to the big leagues!)

Think about what the word *radian* sounds like...well, it sounds like 'radius', right? It turns out that a radian has a close relationship to the radius of a circle

#### So what is a radian then?

**Definition of radian: ** a radian is the measure of an angle that, when drawn a central angle of a circle, intercepts an arc whose length is equal to the length of the radius of the circle.

### Demonstration of Radian of Circle

The video below is an animation of 1 radian. Notice how the length of 1 radius stretches out to a portion of the circle. That portion is 1 radian of the circle.

There is a simple formula to convert radians to degrees. 1Π radian = 180. Therefore you can easily convert from one unit of measure to the other

### Degrees to radians

The general formula for converting from degrees to radians is to simply multiply the number of degree by Π /180°

##### Example 1

Convert 200° into radian measure:

$ 200 ^{\circ} \cdot \frac{\pi }{ 180 ^{\circ}} \\ = \frac{10\pi}{ 9} \text{ radians } = \text{3.49 radians } $

##### Example 2

Convert 120° into radian measure:

$ 120 ^{\circ} \cdot \frac{\pi }{ 180} = \frac{2\pi}{ 3}\text{ radians }= \text{2.09 radians } $

### Radians to degrees

The general formula for converting from radians to degrees to simply multiply the number of degree by 180°/(Π)

##### Example 1

Convert $$ \frac{4}{9} \pi \text{ radians} $$ to degrees

$ \frac{4\pi}{9} \cdot \frac{180 ^{\circ} }{\pi} \\ = \frac{4\pi \cdot 180 ^{\circ} \cdot}{9\pi} =\frac{720 ^{\circ} \pi \cdot}{9\pi} \\ \frac{720 ^{\circ} \cancel{ \color{Red} \pi} \cdot}{9\cancel{ \color{Red} \pi}} \\ = 80 ^{\circ} $

##### Example 2

Convert 1.4 radians into degrees:

$ 1.4 \cdot \frac{180 ^{\circ} }{\pi} \\ =\frac{252 ^{\circ} }{\pi} \approx 80.2^{\circ} $

#### So what's the deal with '$$\pi $$ radians' vs 'radians'?

Let's rephrase the question as follows:

Is there any difference between 5 radians and $$ 5\pi \text{ radians }$$?

Well, let's figure out the answer by converting 5 radians to degrees and $$ 5\pi \text{ radians }$$ to degrees.

If we end up with the same number, then 5 radians and $$ 5\pi \text{ radians }$$ are the same.

$ 5 \cdot \frac{180 ^{\circ} }{\pi} \\ =\frac{900 ^{\circ} }{\pi} \\ \approx 286.4^{\circ} $

$ 5\pi \cdot \frac{180 ^{\circ} }{\pi} \\ =\frac{900 \pi ^{\circ} }{\pi} \\ =\frac{900 \cancel{ \color{Red} \pi} ^{\circ} }{ \cancel{ \color{Red} \pi} } \\ = 900^{\circ} $

**Practice** Converting Degrees between Radians

Use the formula to convert degrees to radians and multiply the degrees by $$ \frac{\pi}{180}$$.

Use the formula to convert radian to degrees and multiply the radian measure by $$ \frac{180}{\pi}$$.

Use the formula to convert radian to degrees and multiply the radian measure by $$ \frac{180}{\pi}$$.

$ \frac{7\pi}{12} \cdot \frac{180 ^{\circ} }{\pi} \\ = \frac{7\pi \cdot180 ^{\circ} }{12\pi} \\ = \frac{7 \cancel{\color{Red}{\pi}} \cdot180 ^{\circ} }{12\cancel{\color{Red}{\pi}} } \\ = \frac{1260 ^{\circ} }{12} = 105 ^{\circ} $

Use the formula to convert radian to degrees and multiply the radian measure by $$ \frac{180}{\pi}$$

$ 14 \cdot \frac{180 ^{\circ} }{\pi} \\ = \frac{ 14 \cdot180 ^{\circ} }{\pi} \\ = \frac{ 2520 ^{\circ} }{\pi} \\ \approx 802.1 ^{\circ} $