Question

A piece of wire 20 cm long is bent into the form of an arc of a circle subtending an angle of 60 degrees at its centre. Then the radius of the circle is
A. \[\dfrac{60}{\pi }cm\]
B. \[\dfrac{120}{\pi }cm\]
C. \[\dfrac{30}{\pi }cm\]
D. \[\dfrac{90}{\pi }cm\]

Answer 1

Hint: Use the arc length formula that is given as: \[arc\,\,length\,\,=\dfrac{\theta }{{{360}^{o}}}\times \left( 2\pi r \right)\]. Here \[\theta \] is the angle subtended at the centre of the circle, r is the radius of the circle. Also, the arc length is the same as the length of the wire.

Complete step-by-step solution -
In the question, we have to find the radius (r) of the circle.
We are given that a piece of wire 20 cm long is bent into the form of an arc of a circle subtending an angle of 60 degrees at its centre. So, here the arc length will be the same as 20 cm, as shown in the figure below.

Next, we know the arc length formula as \[arc\,\,length\,\,=\dfrac{\theta }{{{360}^{o}}}\times \left( 2\pi r \right)\], here \[\theta \] is the angle subtended at the centre of the circle. So we have: \[\theta ={{60}^{o}}\] and \[arc\,\,length\,\,=20\,\,cm\]. So using the formula we will find the radius of the circle, as shown below:
\[\begin{align}
  & \Rightarrow arc\,\,length\,\,=\dfrac{\theta }{{{360}^{o}}}\times \left( 2\pi r \right) \\
 & \Rightarrow 20=\dfrac{{{60}^{o}}}{{{360}^{o}}}\times \left( 2\pi r \right) \\
 & \Rightarrow 20=\dfrac{1}{6}\times \left( 2\pi r \right) \\
 & \Rightarrow 60=\left( \pi r \right) \\
 & \Rightarrow r=\dfrac{60}{\pi } \\
\end{align}\]
So we get the required radius of the circle as \[r=\dfrac{60}{\pi }cm\]. Hence, the correct answer is option A.

Note: Keep in mind that all the lengths are to be in the same unit. For example, if cm is used then all are to be in cm only. Also, the angle \[\theta \] is to be in degrees and not in radians. Sometimes students make mistakes by adding 2r in the arc length which is wrong as we use this approach when there is asked about the segment of a circle.