Radians and degrees are the two different ways to measure angles. In the radian system of angular measurement, the measure of one revolution is 2$\pi$. In many areas of mathematics, radian is considered to be the standard unit of angular measure. The proportionate ratio between the length of an arc and its radius is used to define radian measure. The radian measure of an angle is the length of the arc along the circumference of the unit circle cut off by the angle. |

Relation between the arc length and radius is as follows:

Unit radian is used to denote measurement of angle. Basically, one radian is angle subtended by an arc of unit length that is intercepted on a circle of radius one. Angle which is made by interception of arc of a unit radius circle is referred as unit radian. When we calculate central angle, ratio of arc and radius of circle represents radian measurement of angle. We can define 180 degrees as $\pi$ radians and 90 degrees as $\frac{\pi}{2}$ radians.

The radian measure $\theta$ of the angle is the ratio of the arc length "s" to the radius "r". Formula for radian measure of an angle is as follows:

$\theta$ = $\frac{s}{r}$

where, s is the length of the arc intercepted by the circle.

$\theta$ is measured in radians.

Radian measure $\theta$ of a central angle of a circle is defined as the ratio of the length of the arc the angle subtends, s and the radius of the circle, r.

$\theta$ = $\frac{\text{Length of subtended arc}}{\text{Length of radius}}$

$\theta$ = $\frac{s}{r}$

Where, 'r' is radius of circle and 's' is length of arc subtended by central angle $\theta$.

Mathematical constant $\pi$ can be used to represent an angle in radians. For instance, 90 degrees can be written as $\frac{\pi}{2}$ in radians and 30 degrees can be written as $\frac{\pi}{6}$ in radians.

To convert radians to degrees, we make use of the fact that $\pi$ radians equals one half circle, or 180$^{\circ}$

To convert radians to degrees, multiply by $\frac{180}{\pi}$ Degrees = Radians x $\frac{180}{\pi}$

Given below are few examples in converting radians to degrees.

### Solved Examples

**Question 1:**Convert $\frac{7}{3}$ $\pi$ radians to degrees.

**Solution:**

Degrees = Radians x $\frac{180^{\circ}}{\pi}$

= $\frac{7}{3}$ $\pi$ x $\frac{180^{\circ}}{\pi}$

= $\frac{7\pi \times180^{\circ}}{3\pi}$

= $\frac{1260^{\circ}}{3}$

= 420$^{\circ}$

**Question 2:**Convert $\frac{8}{9}$ $\pi$ radians to degrees.

**Solution:**

Degrees = Radians x $\frac{180^{\circ}}{\pi}$

= $\frac{8}{9}$ $\pi$ x $\frac{180^{\circ}}{\pi}$

= $\frac{8\pi \times 180^{\circ}}{9\pi}$

= $\frac{1440^{\circ}}{9}$

= 160$^{\circ}$

Radians = Degrees x $\frac{\pi}{180}$

Given below are some examples in converting degrees to radians.

### Solved Examples

**Question 1:**Convert 540$^{\circ}$ into radian measure.

**Solution:**

Radians = Degrees x $\frac{\pi}{180}$

= 540$^{\circ}$ x $\frac{\pi}{180}$

= $3\pi$

**Question 2:**Convert 135$^{\circ}$ into radian measure.

**Solution:**

Radians = Degrees x $\frac{\pi}{180}$

= 135$^{\circ}$ x $\frac{\pi}{180}$

= $\frac{3\pi}{4}$