Angles are used to measure the separation between two objects or lines. There are 2 common ways of expressing an angle. The first is degrees, there are 360º in a full circle, so quarter of a circle is 90º, and an eighth of a circle is 45º.
It is convention to measure angles in an anticlockwise direction from the positive 
-axis, therefore negative angles move clockwise.
The second common method of measuring an angle is in radians. A radian is the measure of how far around a circle you have moved. A radian is related to the radius of the circle expressed as a fraction of the circumference. We know that a circle has a circumference of:
(1) 
Taking the radius to be 1, we can simplify this to:
(2) 
So therefore we can see that there are 
radians in a complete circle. As 
is approximately 3.14, we can see that a circle has approximately 6.28 radians. We can use to express the angle in a mathematically precise manor, a half circle has radians and a quarter circle has 
radians.
To convert between radians and degrees we start by knowing that a complete circle has:
(3) 
So to convert from radians to degrees, we can do:
![\[ \begin{split} \label{eq:RadiansToDegrees} \text{Angle}^\circ &= \text{Angle}^r \times \frac{360^\circ}{2 \pi \text{ rad}} \\ &= \text{Angle}^{r} \times \frac{180^\circ}{\pi \text{ rad}} \end{split} \]](https://andy.bettger.net/wp-content/ql-cache/quicklatex.com-3482ff0e551379588c47588f7d2ce100_l3.svg "Rendered by QuickLaTeX.com")
To convert from degrees to radians we use:
![\[ \begin{split} \label{eq:DegreesToRadians} \text{Angle}^{r} &= \text{Angle}^{\circ} \times \frac{2\pi \text{ rad}}{360^\circ} \\ &= \text{Angle}^{\circ} \times \frac{\pi \text{ rad}}{180^\circ} \end{split} \]](https://andy.bettger.net/wp-content/ql-cache/quicklatex.com-cc3668a7250dbe5a0ad023f4761e24c0_l3.svg "Rendered by QuickLaTeX.com")
In astronomy it is also common to measure fractions of angles using fractions of a degree called arcminutes and arcseconds, these are simply 
of a degree for an arcminute and an arcsecond is of an arcminute or 
of a degree. The symbol for arcminutes is the prime 
and for arcseconds double prime 
is used. For example the Moon and Sun as viewed from Earth are approximately half a degree across, which we could say in arcminutes is 
and in arcseconds is 
. We could express this angular dimension fully as 
. This is the degrees minutes and seconds or DMS system.
To convert from DMS to decimal degrees for use in calculations we just divide the arcminutes by 60 and arcseconds by 3600 and add these two results to the whole degree to get the total measurement. Conversely to convert to DMS we use the fraction part of the degrees measurement and multiply it by 60, then take the integer part of the result to give arcminutes, take the fraction part of this result and multiply it by 60 to give arcseconds.
It is also possible to measure even smaller angles using milliarcseconds and microarcseconds which are 
of an arcsecond and of a milliarcsecond respectively. While milliarcseconds and microarcseconds do not have official symbols they are commonly abbreviated to 
and 
respectively.
