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, $\pi =\dfrac{22}{7}$]

& 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{aligned}$$ Thus we got the angle subtended by the arc as $${{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{aligned} & =\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{aligned}$$ Thus we got the area of the sector as $$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.