Question

Calculate the length of the arc of a circle of radius $31.0\,cm$ which subtends an angle of $\dfrac{\pi }{6}$ at the centre.

Answer 1

Hint: Here, we have to calculate the length of the arc of a circle whose radius is $31.0\,cm$ and subtends an angle of $\dfrac{\pi }{6}$ at the centre. So, we will use the formula $l = r \times \theta $ where $l = $ length of arc, $r = $ radius of the circle and $\theta = $ angle subtended at the centre and perform the required calculations to get the result.

Complete step by step solution:
In a circle the length of the arc is directly proportional to the angle covered by the arc, if the angle of arc is $2\pi $ then the length of the arc is $2\pi r$ where $r$ stands for the radius of the circle and is also termed as the proportionality constant.
Here, we have to calculate the length of the arc of a circle whose radius is $31.0\,cm$ and subtends an angle of $\dfrac{\pi }{6}$ at the centre.
So, using the formula $l = r \times \theta $ and putting the value of the given data. We get,
$ \Rightarrow l = 31.0 \times \dfrac{\pi }{6}$
Put the value of $\pi = 3.14$. We get,
$ \Rightarrow l = 31.0 \times \dfrac{{3.14}}{6}$
On multiplying the numbers we get,
$ \Rightarrow l = \dfrac{{97.34}}{6}$
On dividing the numbers we get
$ \Rightarrow l = 16.22\,cm$
Hence, the length of the arc $l = 16.22\,cm$.

Note: In order to solve these types of problems if we have a given the angle in degree measure we must have to convert it first in radian before putting the values in the formulas by using the formula $rad\,measure\, = \dfrac{\pi }{{180}} \times \deg \,measure$ as in the formula the length of the arc is equal to the product of radius and the angles subtended by the arc, so the angle should be in radian measure.