In geometry in math, angles can be measured in radians or degrees. Measuring angles is an important aspect of geometry as most of the geometry and all of trigonometry is based on angle measures. Polar coordinate geometry also involves angle measures. Angle measures are also useful in calculus, in evaluating definite integrals. In short, it is not possible to proceed anywhere beyond algebra in math if we do not understand measuring angles.

The degree measure for angles is introduced to students at an early state, say grade $3$ or $4$. The radian measures are not normally taught before 10th grade. Let us first understand the degree measure.
The central angle of a circle can be divided into $360$ degrees. We use a semicircular protractor to measure and sketch angles to the nearest degree. However it is possible to measure angles in units smaller than degrees also. Just like the smaller unit of meter is centimeter, similar the smaller unit of degree is a minute. There are $60$ minutes in a degree. In other words a single degree can be subdivided into $60$ parts each being one minute. The minute again can be divided to seconds. Again there are $60$ seconds in a minute. The symbol for degree is $^{\circ}$, that for minute is ' and that for seconds is ". Thus if an angle measures d degrees, m minutes and s seconds, it can be written like this:
$d^{\circ}m's"$
The radian angle measure is a real number angle measure. Just like any other real number, the radian angle measure can be written in terms of fractions or decimals. It is however customary to write radian angle measures in term of multiples of $\pi$. We’ll look at why and how is that done in the next section.
We know that a circle can be divided into $360$ degrees. Now if the radius of this circle is $1$ unit, then its circumference is given by $2 \pi\ \times\ 1$ = $2 \pi$. This $2 \pi$ is a real number. Thus we say that the angle at the center of a full circle is $360^{\circ}$ or $2 \pi^{R}$. The $^{R}$ is used to indicate that the given angle is in radians.
If the angle given to us is in complete degrees, then it can be converted to radians as follows:
We know that,
$360^{\circ}$ = $2 \pi^{R}$
We want to find what would $x^{\circ}$ equal in radians. Then using proportions we have:
$x^{\circ}$ = $(\frac{2 \pi \times x}{360})^{R}$
Simplifying that we have:
$x^{\circ}$ = $\frac{\pi}{180}$ $x^{R}$
However, if the angle given to us is not just in degrees, but in degrees, minutes and seconds, we need to first convert it to just degrees and only then we can further convert it to radians. This part will be more clear once we do examples at the end of the article.
If the angle given to us is in radians then, it can be converted to degrees as follows:
We again now know that:
$2 \pi^{R}$ = $360^{\circ}$
Therefore $y^{R}$ would equal,
$y^{R}$ = $(\frac{y \times 360}{2 \pi})^{\circ}$
Simplifying that we have:
$y^{R}$ = $(\frac{180}{p}y)^{\circ}$
In simpler words, to convert from degrees to radians we multiply by $\frac{\pi}{180}$ and to convert back from radians to degrees we divide by $\frac{\pi}{180^{\circ}}$.
From our grade $6$ knowledge we know that the circumference of a circle is given by the formula:
$C$ = $2 \pi r$
Now, we also know that the angle subtended by the complete circle at the centre is full $360$ degrees. Thus if we want to find the length of an arc of a circle that subtends and angle $\theta^{°}$ at the centre, then we can do that by using proportions as follows:
$360^{\circ} \rightarrow\ 2 \pi r$
$\therefore\ \theta^{\circ} \rightarrow$ $(\frac{\theta\ \times\ 2 \pi r}{360})$
The above formula gives length of arc of a circle if the angle theta is given to us in degrees.
Now suppose the subtended angle is given to us in radians instead. Then, we know that angle subtended by complete circle is $2 \pi$ radians, therefore our proportion would look like this:
$2 \pi^{R} \rightarrow\ 2 \pi r$
$\therefore\ \theta^{R}\ \rightarrow\ 2 \pi r\ \times$ $\frac{\theta}{2 \pi}$ = $r \theta$
In other words, the length of an arc subtending an angle of $\theta^{R}$ at the centre is $r \theta$.
Example 1:
Convert from degrees to radians the angle measure: $20^{\circ}15'24"$
Solution:
We know that
$20^{\circ}$ = $20^{\circ}$
We also know that,
$60'$ = $1^{\circ}$
Therefore,
$15'$ = $15 \times$ $\frac{1}{60}$ = $0.25^{\circ}$
Similarly we also know that:
$60"$ = $1'$
Therefore,
$(60 \times 60)"$ = $60'$ = $1^{\circ}$
$3600"$ = $1^{\circ}$
Thus,
$24"$ = $24 \times$ $\frac{1}{3600}$ = $0.00667^{\circ}$
Thus,
$20^{\circ}15'24"$ = $20 + 0.25 + 0.00667$ = $20.25667^{\circ}$
This is how we convert from degrees, minutes and seconds to decimal degrees.
Now the conversion factor for degrees to radians is multiplication by $\frac{p}{180}$. Therefore,
$20.25667^{\circ}$ = $20.25667 \times$ $\frac{p}{180}$ = $0.3535^{R} \leftarrow$ Answer!
Example 2:
Convert the following angle measure from radians to degrees: $\frac{p}{15}$ radians.
Solution:
We know that to convert from radians to degrees we need to divide by $\frac{p}{180}$. Thus,
$\frac{\pi^{R}}{15}$ = $\frac{\frac{\pi}{15}}{\frac{\pi}{180}}$ = $\frac{\pi}{15}$ $\times$ $\frac{180}{\pi}$ = $\frac{180}{15}$ = $12^{\circ} \leftarrow$ Answer!
Example 3:
Find the arc length of a circle of radius $8$ cm if the arc subtends an angle or $\frac{\pi}{6}$ radians at the centre.
Solution:
We learned a little while ago that the length of an arc of a circle when the angle subtended is in radians is:
$L$ = $r \theta$
For this problem,
$r$ = $8$ cm
And
$\theta$ = $\frac{\pi}{6}$
Therefore the arc length would be:
$L$ = $8 \times$ $\frac{\pi}{6}$ = $4.19$ cm $\leftarrow$ Answer!