Question

The angle of a sector in a given circle is 40 degree and the area of the sector is equal to \[20c{m^2}\]. Calculate the arc length of the sector.
A 6.5 cm
B 7.4 cm
C 5.3 cm
D None of these

Answer 1

Hint: In this problem, first we need to find the radius of the sector using the formula of the area of the sector. Now, substitute the value of radius in the arc length formula to obtain the arc of the sector.

Complete step-by-step answer:
Consider, the angle of the sector be \[\theta\], radius of the sector be\[r\].
The formula for the area \[A\] of a sector is shown below.
\[A = \left( {\dfrac{\theta }{{360}}} \right)\pi {r^2}\]
Since, the area of the sector is given as \[20c{m^2}\], and angle of the sector is given as 40 degree, substitute, 20 for \[A\] and 40 for \[\theta \] in above equation.
\[\begin{gathered}
  \,\,\,\,\,20 = \left( {\dfrac{{40}}{{360}}} \right)\pi {r^2} \\
   \Rightarrow 20 = \left( {\dfrac{1}{9}} \right)\pi {r^2} \\
   \Rightarrow \pi {r^2} = 180 \\
   \Rightarrow r = \sqrt {\dfrac{{180}}{\pi }} \\
\end{gathered}\]
Now, the formula for the arc length \[S\] is shown below.
\[S = \left( {\dfrac{\theta }{{180}}} \right)\pi r\]

Substitute 40 for \[\theta \] and \[\sqrt {\dfrac{{180}}{\pi }}\] for \[r\] in the above formula to obtain the arc length.
\[\begin{gathered}
  \,\,\,\,\,\,S = \left( {\dfrac{{40}}{{180}}} \right)\pi \sqrt {\dfrac{{180}}{\pi }} \\
   \Rightarrow S = \left( {\dfrac{2}{9}} \right)\sqrt {180\pi } \\
   \Rightarrow S = \left( {\dfrac{2}{3}} \right)\sqrt {20\pi } \\
   \Rightarrow S = 5.29 \\
   \Rightarrow S \simeq 5.3 \\
\end{gathered}\]
Thus, the arc length of the sector is 5.3 cm, hence, option (C) is the correct answer.

Note: Arc is defined as the product of the radius and angle of the sector (in radian). In other words, the angle of a sector is defined as the ratio of the arc of the sector to radius of the sector.