Question

In a circle of radius 21cm, an arc subtends an angle of $60{}^\circ $at the centre. Find
(i) The length of the arc
(ii) Area of sector.

Answer 1

- Hint: Use the formula $\theta =\dfrac{l}{r}$ where $\theta $ = angle subtended by arc, l = length of the arc and r = radius of the circle. For calculating the length of the arc as ‘$\theta $’ and ‘r’ are given in question.
For finding the area of the sector use formula, Area of the sector $=\dfrac{1}{2}\times \theta \times {{r}^{2}}$.

Complete step-by-step solution -

According to question, radius of the circle = 21 cm and angle subtended by arc = $60{}^\circ $.
We know, $\theta =\dfrac{l}{r}$.
First we will need to change ‘$\theta $’ and ‘r’ in their SI unit. SI unit of radius is meter and angle is radian.
As, 1m = 100cm and \[1{}^\circ =\dfrac{\pi }{180}radian,\]
 \[\begin{align}
  & radius=21cm=\dfrac{21}{100}=0.21m \\
 & Angle\ subtended=60{}^\circ =\left( 60\times \dfrac{\pi }{180} \right)radian=\dfrac{\pi }{3} \\
\end{align}\]
Now, putting these values in the formula,
$\begin{align}
  & \theta =\dfrac{l}{r} \\
 & \dfrac{\pi }{3}=\dfrac{l}{0.21} \\
\end{align}$
Multiplying both sides by 0.21, we will get,
$\begin{align}
  & \dfrac{\pi }{3}\times 0.21=\dfrac{l}{0.21}\times 0.21 \\
 & \Rightarrow \pi \times 0.07=l \\
\end{align}$
Using $\pi =3.14$, we will get,
$\begin{align}
  & \Rightarrow l=\left( 3.14\times 0.07 \right)m \\
 & \Rightarrow l=0.2198m \\
 & \Rightarrow l=\left( 0.2198\times 100 \right)cm\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left[ As,\ 1m=100cm \right] \\
 & \Rightarrow l=21.98cm \\
\end{align}$
Hence the required length of the arc = 21.98cm.
Now, we have to calculate area of the sector;
We know, Area of the sector $=\dfrac{1}{2}\times \theta \times {{r}^{2}}$.
Putting all the values in SI unit;
$\begin{align}
  & Area=\dfrac{1}{2}\times \dfrac{\pi }{3}\times \left( 0.21 \right)\times \left( 0.21 \right) \\
 & =\dfrac{1}{2}\times \dfrac{3.14}{3}\times \left( 0.21 \right)\times \left( 0.21 \right) \\
 & =0.023079{{m}^{2}} \\
\end{align}$
We know $1{{m}^{2}}={{10}^{4}}c{{m}^{2}}$
$\begin{align}
  & \Rightarrow Area=\left( 0.023079\times {{10}^{4}} \right)c{{m}^{2}} \\
 & \Rightarrow Area=230.79c{{m}^{2}} \\
\end{align}$
Hence, the required area of the sector $=230.79c{{m}^{2}}$.

Note: In the formulas of length of the arc and area of the sector, use all the quantities in SI unit. Don’t forget to convert angle into radian. SI unit of angle is radian.