Question

How do you find arc length of an arc that subtends a central angle of 60 degrees in a circle with radius 25m?

Answer 1

Hint: This type of problem is based on the concept of finding arc length of a circle. First, we have to convert 60 degrees to radian measure, that is, \[2\pi =360\] degree. Then, substitute the obtained values in the formula \[l=r\times \theta \], where l is the length of the arc and \[\theta \] is the angle subtended in radian measure. Here, r= 25 cm and \[\theta =\dfrac{\pi }{3}\]. Substitute these values in the formula. Make necessary calculations and find the value of l in cm.

Complete step by step answer:
According to the question, we are asked to find the length of the arc that subtends a central angle of 60 degrees in a circle with radius 25m.
We have been given the radius is 25 cm and angle is 60 degrees.
We first have to consider the angle 60 degrees.
First, we have to convert the degree into radian.
We know that \[2\pi =360\] degree.
Divide the whole equation by 6. We get,
\[\dfrac{2\pi }{6}=\dfrac{360}{6}\]degree
 On further simplification, we get,
\[\Rightarrow \dfrac{2\pi }{6}=\dfrac{6\times 60}{6}\]
\[\Rightarrow \dfrac{\pi }{3}={{60}^{\circ }}\]
\[\therefore \theta =\dfrac{\pi }{3}\] ---------(1)
We know that, the formula to find the length of the arc when angle and radius are given, is
\[l=r\times \theta \], ------------(2)
where l is the length of the arc and \[\theta \] is the angle subtended.
Given in the question that r=25cm.
And from (1), we get \[\theta =\dfrac{\pi }{3}\].
Let us substitute these values in formula.
We get,
\[l=25\times \dfrac{\pi }{3}\]
\[\Rightarrow l=\dfrac{25\pi }{3}\]
We know that the value of \[\pi \] is \[\pi =3.14\].
Therefore,
\[\Rightarrow l=\dfrac{25\left( 3.14 \right)}{3}\]
On further simplification, we get,
\[\Rightarrow l=\dfrac{78.4}{3}\]
\[\therefore l=26.1666\]cm
 Rounding up the final answer, we get,
\[l=26.16\]cm
Hence, the arc length of an arc that subtends a central angle of 60 degrees in a circle with radius 25m is 26.16cm.

Note: Whenever you get this type of problem, we should always try to make the necessary calculations in the given equation to get the final answer. We should avoid calculation mistakes based on sign conventions. We should not forget to convert degree into radiant. We should always consider the value of \[\pi \] as 3.14 approximately. The final solution can also be written as \[\dfrac{25\pi }{3}\].