Question

What is the length of an arc of a circle with a radius of \[5\] . If it subtends an angle of \[{60^o}\] at the centre ?
\[(1)\] \[3.14\]
\[(2)\] \[5.24\]
\[(3)\] \[10.48\]
\[(4)\] \[2.62\]

Answer 1

Hint: We have to find the length of an arc of a circle which is formed when an angle of \[{60^o}\] is subtended at the centre of the circle and has a radius of \[5\] . We solve this question using the concept of length of arc of a circle . We should also have the knowledge about the relation between the degree and radian measure of a value . Using the formula for the arc length of the circle and converting the value of angle given in degrees to radians we can simply calculate the value of the arc length of the circle .

Complete step-by-step solution:
Given :
The value of \[radius\left( r \right) = 5\]
The angle subtended at the centre \[(\theta ) = {60^o}\]
Now , as the value of \[\theta \] is in degrees .
So , we would use the formula for arc length of the circle where the angle subtended at the centre is in degrees .
The formula is as given below :
\[L = \dfrac{\theta }{{{{360}^o}}} \times 2\pi r\]
Where \[L\] is the length of the arc , \[\theta \] is the measure of the angle subtended at the centre in degrees and \[r\] is the radius of the circle
Putting , the values in the formula we get the value of length of arc of the circle
\[L = \dfrac{{{{60}^o}}}{{{{360}^o}}} \times 2\pi \times 5\]
On solving , we get the value of arc length
\[L = 5.24units\]
(As the units of measurement of the radius are not given we consider them to be constant and state them as units)
Hence , the length of arc of the circle is \[5.24units\] .
Thus , the correct option is \[(2)\] .

Note: Now , we also know that the formula for arc length of a circle is given as :
\[L = r \times {\theta ^c}\]
Where \[L\] is the length of arc of the circle , \[r\] is the radius of the circle and \[\theta \] is the value of angle subtended at the centre of the circle .
[Here the value of \[\theta \] is in radians . So , we have to convert the value of theta given in degrees to radians or we will accordingly change our formula for the arc length.