Angles are commonly used in our day to day life. The two types of units of measurement of angle are radians and degree. Most often, we use the degree for measuring and denoting angles. Degree and radian can also be converted into each other. Let’s learn about both the units of angles in detail.

## Definition

A degree is defined as $\frac{1}{360}$th of a complete circle. The symbol used to denote degree is a small circle at the top right of the value $^o$. Smaller units of degree and are minutes and seconds where 1$^o$ = 60 minutes and 1 minute = 60 seconds.

Radian is considered to be the SI unit of measurement of angle. One radian is defined as the angle drawn at the center of the circle by an arc whose length is equal to that of the measure of the radius of the circle.

## Converting Degrees to Radians

The relation between the degree and radian is as follows:

$\pi$ radians = 180 degrees

So, $1^o$ = $\frac{\pi}{180}$ radians = $\frac{3.14}{180}$ radians = $0.01746$ radians

The angle $\theta$ in radians is equal to $\theta$ in degrees times pi divided by 180 degrees.

$\theta_{radians}$ =$\theta_{degress}$ $\times$ $\frac{\pi}{180}$
Example:

Convert 60 degrees angle to radians:

Solution - We know that

$\theta_{radians}$ = $\theta_{degrees}$ $\times$ $\frac{\pi}{180}$

$\theta_{radians}$ = $60\ \times$ $\frac{\pi}{180}$ radians = $\frac{\pi}{3}$ radians = $\frac{3.14}{3}$ radians = 1.0467 radians

Conversion of standard angles lying on the unit circle:

Degrees Radians
0$^o$ 0 radians
30$^o$ $30\ \times$$\frac{\pi}{180} = \frac{\pi}{6} radians = \frac{3.14}{6} = 0.52radians 45^o 45\ \times \frac{\pi}{180} = \frac{\pi}{4} radians = \frac{3.14}{4} = 0.79 radians 60^o 60\ \times \frac{\pi}{180} = \frac{\pi}{3} radians = \frac{3.14}{3} = 1.047 radians 90^o 90\ \times \frac{\pi}{180} = \frac{\pi}{2} radians = \frac{3.14}{2} = 1.57 radians 120^o 120\ \times \frac{\pi}{180} = 2$$\frac{\pi}{3}$ radians = $2\ \times$ $\frac{3.14}{3}$ = 2.09 radians
135$^o$ $135\ \times$ $\frac{\pi}{180}$ = $3$$\frac{\pi}{4} radians = 3\ \times 3.\frac{14}{4} = 2.36 radians 150^o 150\ \times \frac{\pi}{180} = 5$$\frac{\pi}{6}$ radians = $5\ \times$ $\frac{3.14}{6}$ = 2.62 radians
180$^o$ $180\ \times$ $\frac{\pi}{180}$ = $\pi$ radians = 3.14 radians
270$^o$ $270\ \times$ $\frac{\pi}{180}$ = $3$$\frac{\pi}{2} radians = 3\ \times \frac{3.14}{2} = 4.71 radians 360^o 360\ \times \frac{\pi}{180} = 2\ \pi radians = 2\ \times\ 3.14 = 6.18 radians ## Converting Radians to Degrees The relation between the degree and radian is as follows: 180 degrees = \pi radians So, 1 radian = \frac{180^o}{\pi} = \frac{3.14}{180} radians = 57.325 radians The angle \theta in degrees is equal to \theta in radians times 180 degrees divided by \PI constant: \theta_{degrees} = \theta_{radians}\ \times \frac{180^o}{\pi} Example: Convert 3 radians angle to degrees: Solution - We know that \theta_{degrees} = \theta_{radians}\ \times \frac{180^o}{\pi} \theta_{degrees} = 3\ \times \frac{180^o}{\pi} degrees = 3\ \times \frac{180^o}{3.14} degrees = 171.97 degrees Conversion of standard angles lying on the unit circle: Radians Degrees 0 radians 0 degrees \frac{\pi}{6} radians \frac{\pi}{6}$$\times$ $\frac{180^o}{\pi}$ = 30 degrees
$\frac{\pi}{4}$ radians $\frac{\pi}{4}$ $\times$ $\frac{180^o}{\pi}$ = 45 degrees
$\frac{\pi}{3}$ radians $\frac{\pi}{3}$ $\times$ $\frac{180^o}{\pi}$ = 60 degrees
$\frac{\pi}{2}$ radians $\frac{\pi}{2}$ $\times$ $\frac{180^o}{\pi}$ = 90 degrees
2$\frac{\pi}{3}$ radians 2$\frac{\pi}{3}$ $\times$ $\frac{180^o}{\pi}$ = 120 degrees
3$\frac{\pi}{4}$ radians 3$\frac{\pi}{4}$ $\times$ $\frac{180^o}{\pi}$ = 135 degrees
5$\frac{\pi}{6}$ radians 5$\frac{\pi}{6}$ $\times$ $\frac{180^o}{\pi}$ = 150 degrees
$\pi$ radians $\pi\ \times$ $\frac{180^o}{\pi}$ = 180 degrees
3$\frac{\pi}{2}$ radians 3$\frac{\pi}{2}$ $\times$ $\frac{180^o}{\pi}$ = 270 degrees
2$\pi$ radians 2$\pi\ \times$ $\frac{180^o}{\pi}$ = 360 degrees

## Decimal Degrees

Decimal degrees are used to represent longitude and latitude geographic coordinates as fractions in decimals. Its main application is in web mapping as in GPS devices and open-street map. They are an alternate form of degree, minutes, and seconds (DMS).

One degree $1^o$ = $60’$ (60 minutes)

One minute $1’$ = $60’’$ (60 seconds)

i)    Stepwise illustration of converting decimal degrees (DD) to decimals minutes and seconds (DMS) with the help of an example:

Example: Convert decimal degrees 8.23456 (DD) into degrees, minutes, and seconds (DMS)

Solution – Step 1: Subtract the whole degrees.

8.23456 – 8 = 0.23456 (8 is the whole number)

Step 2: Convert the fractional degree into minutes by multiplying the decimal part of the degree by 60. The whole number that we get is the minutes

$0.23456$ $\times$ $60’$ per degree = $14.0736’$

Step 3: Similarly, we subtract the whole minutes from the fractional minutes we found out

$14.0726’$ $-\ 14$ = 0$.0726’$ (14 is the whole minutes)

Step 4: Convert the fractional minutes into seconds by multiplying the decimal part of the minutes by 60. The whole number that we get is the seconds. The decimal part of the second is no more converted and stays as the decimal seconds.

$0.0726\ \times\ 60$ = $4.356’’$

So, the decimal, minutes, seconds that we get is $8^o\ 14’\ 4.356’’$
ii)    Stepwise illustration of converting decimals minutes and seconds (DMS) to decimal degrees (DD) with the help of an example:

Example: Convert decimal minutes and seconds $25^o\ 8’\ 10’’$ (DMS) into decimal degrees (DD)

Solution – Step 1: Convert the seconds to minutes. We know that 60’’ is equal to 1’. Therefore, 10’’ is equal t 10/60 = 0.167’ (minutes)

Step 2: Add the decimal minutes to the whole minutes. The whole minutes is 8 given and we found out the decimal minutes to be 0.167.  So, the total minutes would be $8\ +\ 0.167$ = $8.167‘$

Step 3: Convert the minutes to degrees. We know that 60’ is equal to $1^o$. Therefore, $8.167’$ is equal to $\frac{8.167}{60}$ = $0.1361^o$

Step 4: Add the decimal degrees to the whole degrees. The whole degree is $25$ and the decimal degrees is $0.1361$. So, the total degrees is $25\ +\ 0.1361$ = $25.1361^o$

So, the decimal degree that we get is $25.1361^o$ (DD).

## Examples

1)    Convert 90 degrees angle to radians:

Solution - We know that

$\theta_{radians}$ = $\theta_{degrees}\ \times$ $\frac{\pi}{180}$

Tradians = $90\ \times$ $\frac{\pi}{180}$ radians = $\frac{\pi}{2}$radians = $\frac{3.14}{2}$ radians = $1.57$ radians
2)    Convert 5 radians angle to degrees:

Solution - We know that

$\theta_{degrees}$ = $\theta_{radians}\ \times$ $\frac{180^o}{3.14}$

$\theta_{degrees}$ = $5\ \times$ $\frac{180^o}{\pi}$ degrees = $5\ \times$ $\frac{180^o}{3.14}$ degrees = $286.62$ degrees
3)    Convert decimal minutes and seconds $2^o\ 36’\ 48’’$ (DMS) into decimal degrees (DD)

Solution – First, we convert the seconds into minutes by dividing by $60$, $\frac{48}{60}$ = $0.8$. Then, add the decimal minute’s to the whole minutes $36$ to get $2^o\ 36.8‘$.

Finally, convert the decimal minutes into degrees by dividing by $60$, $\frac{36.8}{60}$ = $0.613$ and add it to the whole degrees getting $2.613^o$ as the answer.
4)    Convert decimal degrees $7.567$ (DD) into degrees, minutes, and seconds (DMS)

Solution – First we multiply the decimal value with $60$ to get the minutes. $0.567\ \times\ 60$ = $34.02$. The whole number we get is the minutes and then convert the decimal minutes into seconds by again multiplying by $60$. That is the seconds we get $0.02\ \times\ 60$ = $1.2’’$. Therefore, the degrees, minutes and seconds that we get is $7^o\ 34’\ 1.2’’$.