Question

Find the length of an arc of a circle of radius 5 cm subtending a central angle measuring ${{15}^{\circ }}$.

Answer 1

Hint:At first convert the given angle in degree to radian by multiplying it with $\dfrac{\pi }{180}$. Then use the formula $s=r\theta$, where s is length of arc, r is radius and $\theta$ is angle in radian.

Complete step-by-step answer:
In the question we are given measures of radius and central angle which are 5 cm and ${{15}^{\circ }}$ and from this we have to find the length of the arc.
Before proceeding we will first briefly say something about radian.
The radian is an S.I. unit for measuring angles and is the standard unit of angular measure used in areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees.
Radian describes the plain angle subtended by a circular arc as the length of arc divided by radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the magnitude in radians of such a subtend angle is equal to the ratio of the arc length to the radius of circle; that is $\theta \ =\ \dfrac{s}{r}$, where $\theta$ is the subtended angle in radians, s is arc length and r is radius .

Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians that is $s=r\theta$.
Now to find the length of the arc we will use formula $s=r\theta$ where r is radius and $\theta$ is angle in radius and s is the length of arc.
We are given value of angle in degree so we can convert by multiplying it by $\left( \dfrac{\pi }{180} \right)$ so we get $\theta$ in radians as,
$5\times \dfrac{\pi }{180}$ which is equal to $\dfrac{\pi }{12}$.
Hence, using the formula $s=r\theta$ we get,
$s=5\times \dfrac{\pi }{12}$
Which is equal to $\dfrac{5\pi }{12}\ cm$.
So, the length of the arc is $\dfrac{5\pi }{12}\ cm$.

Note: Students generally misunderstand the quantity of $\theta$. Generally, most students have confusion that ‘$\theta$’ in the question is in degree or in radian. So, they should clearly know that the value of $\theta$ is in radian.Students should remember to convert from degree to radian one should multiply by $\dfrac{\pi }{180}$ to get the value in radians and to convert from radian to degree one should multiply by $\dfrac{180 }{\pi}$ to get the value in degrees.