## Key Questions

• Since

${180}^{\circ} = \pi$ radians,

if you want to convert $x$ degrees to radians, then

$x \times \frac{\pi}{180}$ radians,

and if you want to convert $x$ radians to degrees, then

$x \times \frac{180}{\pi}$ degrees

I hope that this was helpful.

$\text{see explanation}$

#### Explanation:

$\text{to convert from "color(blue)"radians to degrees}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\textcolor{b l a c k}{\text{degrees "="radians } \times {180}^{\circ} / \pi}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{to convert from "color(blue)"degrees to radians}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\text{radians "="degrees } \times \frac{\pi}{180} ^ \circ} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

• Imagine a circle and a central angle in it. If the length of an arc that this angle cuts off the circle equals to its radius, then, by definition, this angle's measure is 1 radian. If an angle is twice as big, the arc it cuts off the circle will be twice as long and the measure of this angle will be 2 radians. So, the ratio between an arc and a radius is a measure of a central angle in radians.

For this definition of the angle's measure in radians to be logically correct, it must be independent of a circle.
Indeed, if we increase the radius while leaving the central angle the same, the bigger arc that our angle cuts from a bigger circle will still be in the same proportion to a bigger radius because of similarity, and our measure of an angle will be the same and independent of a circle.

Since the length of a circumference of a circle equals to its radius multiplied by $2 \pi$, the full angle of ${360}^{0}$ equals to $2 \pi$ radians.

From this we can derive other equivalencies between degrees and radians:

${30}^{0} = \frac{\pi}{6}$
${45}^{0} = \frac{\pi}{4}$
${60}^{0} = \frac{\pi}{3}$
${90}^{0} = \frac{\pi}{2}$
${180}^{0} = \pi$
${270}^{0} = 3 \frac{\pi}{2}$
${360}^{0} = 2 \pi$