An angle is formed on a fold or in a corner. How to measure it? What is its unit? There are two units of measurement of the angles. They are radian and degree. In this chapter, we are going understand both of them along with the conversion between them.

Radian is defined as the standard International unit of measurement of angle. It is the angle subtended by an arc of a circle at the center, provided that the arc length is equal to the radius of the circle. Radian measure of an angle is a ratio of arc subtended at the center of the circle to the radius of the same circle. A circle has $2Pi$ radians. A full circle has $360^{\circ}$ . So $2Pi$ radian is equal to $360^{\circ}$. Therefore, a radian would be equivalent to about $\frac{360}{2Pi}$ degrees or $57.296^{\circ}$

Degree is defined as a unit of measurement of angle. It is known that the total degrees in a complete circle are $360$ degrees. The symbol to denote degrees is a small empty circle at the top right of the value of the angle. One degree is one by $360th$ part of a full circle. The relation between a degree and the SI unit of angle radian is “$360^{\circ}$ equals $2Pi$ radian”. So, one degree is $\frac{2Pi}{360}$ radian. There are smaller units than a degree for measurement of angles, they are minutes and seconds. One degree is equal to 60 minutes and one minute is equal to $60$ seconds.
Radians and degrees are both units of measurement of an angle in geometry. Conversion from radian to degree can be derived from the relation between them. It is known that pi radian is equal to $180^{\circ}$. Therefore, one radian is equal to $\frac{180}{pi}$ degrees. Therefore, a number of degrees will be given radian multiplied by $\frac{180}{pi}$. It is basically the product of a number of half circles and $180^{\circ}$. A number of half circle is given by radian divided by Pi.

Formula for conversion from radian to degree is as follows:

$Degree$ = $\frac{Radian}{Pi}$ $\times\ 180$

Alternate formula that could be applied for the conversion of degree to radian is as follows:

$Degree$ = $Radian\ \times\ 57.296^{\circ}$
Radian is the standard international unit in geometry.

The other unit of measurement of an angle in geometry is degree. They both are related to each other and could be converted from radian to degree and from degree to radian. A complete circle has $360^{\circ}$ or $2Pi$ radian in total. So, one degree will be $\frac{2Pi}{360}$ or $\frac{pi}{180}$ radians. Therefore, a number of radians will be the given degree multiplied by $\frac{Pi}{180}$ radian.

The angle, say beta, in radian, is equal to angle beta in degrees times pi divided by $180^{\circ}$.

Formula for conversion from degrees to radians is as follows:

$Radians$ = $Degrees\ \times$ $\frac{Pi}{180}$

Alternate formula that could be applied for the conversion of degree to radian is as follows:

$Radian$ = $Degree\ \times\ 0.0175$ radian

Few of the primary differences between radian and degree are as follows:


Radian is the standard international unit of measurement of an angle in geometry. The degree is not the standard international unit of measurement of an angle in geometry.

Radian is numerically equal to the length of the respective arc in a unit circle. The degree is the angle subtended at the center of a circle by an arc.

In the relation between radian and degree, one radian is equal to $57.3^{\circ}$. In the conversion from degree to radian, one degree is equal to $0.0175$ radians.

In the conversion from radian to degree, the given radian to be converted is multiplied with constant pi divided by $180$. In the conversion from degree to radian, the given degree to be converted is o be multiplied with $180$ and the product is divided by constant pi. The value of pi rounded to two decimal places is $3.14$.

In a full circle, the total angle in radian is $2Pi$. In a complete circle, the total angle in degree is $360^{\circ}$.

Radian measures the distance; it is the ratio of arc covered to radius of the same circle. Degree measures angle covered by the part of the circumference at the midpoint of the circle by the radius of the circle.
Example 1: 

Convert $120^{\circ}$ to radian

Solution: 

Step 1: Pen down the angle in degrees to be converted to radians $120^{\circ}$.

Step 2: Multiply the number of degrees to be converted by $\frac{pi}{180}$.

           $120^{\circ}\ \times$ $\frac{p}{180}$

Step 3: Carry out the multiplication process, which is finding out the product.

             $120^{\circ}\ \times$ $\frac{p}{180}$ = $120$ $\frac{p}{180}$

Step 4: Simplify to the lowest term of the fraction. As both $120$ and $180$ are divisible by $60$, both the numerator and denominator is divided               by $60$ to bring it to the lowest term.

           $\frac{(\frac{120 p}{180})}{(\frac{60}{60})}$ = $2$ $\frac{p}{3}$

Therefore, $120^{\circ}$ converted to radian is $2$ $\frac{p}{3}$.
Example 2: 

Convert $\frac{Pi}{3}$ radian to degrees.

Solution: 

Step 1: Pen down the angle in radian to be converted to degrees.

            $\frac{Pi}{3}$ radian

Step 2: Multiply the number of degrees to be converted by $\frac{180}{pi}$.

           $\frac{Pi}{3}$ $\times$ $\frac{180}{pi}$

Step 3: Carry out the multiplication process, which is finding out the product.

            $\frac{Pi}{3}$ $\times$ $\frac{180}{pi}$ = $60^{\circ}$

Therefore, $\frac{pi}{3}$ radian converted to degrees gives us $60^{\circ}$.
Example 3: 

Convert $45$ degrees into radian and $\frac{Pi}{6}$ radian into degrees.

Solution: 

Degree to radian conversion is done by multiplying with $\frac{Pi}{180}$. So, $45^{\circ}$ would be $45\ \times$ $\frac{Pi}{180}$ = $\frac{Pi}{4}$ radian.

Radian to degree conversion is found out by multiplication with $\frac{180}{Pi}$. So, $\frac{Pi}{6}$ radian would be $\frac{Pi}{6}$ $\times$ $\frac{180}{Pi}$ = $30^{\circ}$.