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Radian Measure

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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.

The abbreviation “rad” is used for the representation of radian measures. For instance, an angle of 15 radian can be written as “15 rad”.

There are also several advantages in measuring angles in radians such as in calculus and also other branches of mathematics which are beyond practical geometry.

Definition of Radian

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Radian is the measure of angle. It describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is equivalent to $\frac{180}{\pi}$ degrees, where $\pi$ is constant and 2$\pi$ radians is equal to 360 degrees. The length of subtended arc is always proportional to the radius of the circle.

Relation between the arc length and radius is as follows:
Measure of Angle in Radian = $\frac{\text{Arc Length}}{\text{Radius}}$

Radian Measure of An Angle

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Angle is defined as the inclination of one ray over other ray and is used in geometry to describe inclination. There are two units which are used to measure angles: one is degree and second is radian. Generally, we use unit "degree" to represent angle’s measurement. Let us now see radian measure of an angle.

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

Radian Measure of Central Angle

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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.

Converting Radians To Degrees

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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:
Formula to convert radians to degrees is
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:
Formula to convert radians to degrees is
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$^{\circ}$

Converting Degrees To Radians

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To convert degrees to radians, multiply by $\frac{\pi}{180}$.
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:
Formula to convert degree to radians is
Radians = Degrees x $\frac{\pi}{180}$
= 540$^{\circ}$ x $\frac{\pi}{180}$
= $3\pi$

Question 2: Convert 135$^{\circ}$ into radian measure.
Solution:
Formula to convert degree to radians is
Radians = Degrees x $\frac{\pi}{180}$
= 135$^{\circ}$ x $\frac{\pi}{180}$ 
= $\frac{3\pi}{4}$