Question

Find the area of the sector whose arc length and radius are 10cm and 5cm respectively.

Answer 1

Hint: From the formula of arc length, substitute the length and radius and find the value of \[\theta \]. Thus find the area of sector by substituting value of \[\theta \] and radius in the formula.

Complete step-by-step answer:

We have been given arc length and radius of the sector of a circle.
The arc length of the sector = 10cm.
Similarly, the radius of the sector = 5cm.
We know that arc length is the distance along the arc or circumference of a circle.
The length of an arc subtending on angle, \[\theta =\dfrac{\theta }{360}\times 2\pi r\].
\[\therefore \] Thus arc length \[=2\pi r\left( \dfrac{\theta }{360} \right)\].
\[10=2\pi \times 5\left( \dfrac{\theta }{360} \right)\] [Take, \[\pi =\dfrac{22}{7}\]]
\[\begin{align}
  & 10=2\times \dfrac{22}{7}\times 5\times \dfrac{\theta }{360} \\
 & \therefore \theta =\dfrac{10\times 360\times 7}{2\times 22\times 5}=\dfrac{360\times 7}{22} \\
 & \theta =\dfrac{180\times 7}{11}={{114.45}^{\circ }} \\
\end{align}\]
Thus we got the angle subtended by the arc as \[{{114.45}^{\circ }}\].
Now we need to find the area of the sector of the circle.
We know the formula to find the area of sector \[=\dfrac{\theta }{360}\times \pi {{r}^{2}}\].
\[\therefore \] Area of sector \[=\dfrac{\theta }{360}\times \pi {{r}^{2}}\], we got \[\theta ={{114.45}^{\circ }}\].
                                \[\begin{align}
  & =\dfrac{{{114.45}^{\circ }}}{360}\times \dfrac{22}{7}\times {{\left( 5 \right)}^{2}} \\
 & =\dfrac{{{114.45}^{\circ }}}{360}\times \dfrac{22}{7}\times 5\times 5 \\
 & =24.97\approx 25c{{m}^{2}} \\
\end{align}\]
Thus we got the area of the sector as \[25c{{m}^{2}}\].

Note: The sector of a circle can also be said as a pizza slice. A circular sector or circle’s sector is the portion of a disk enclosed by 2 radii and an arc. Thus you need to know the basic formula considering sectors to solve similar problems.