Question

The perimeter of the sector of a circle, of area 25$\pi $sq. cm. is 20 cm. Find the area of the sector.

Answer 1

Hint: We will first find the radius of the circle by comparing the area 25$\pi $ with the formula of the area of the circle $\pi r^2$. Then, we will calculate the arc length s of the sector using the perimeter of the sector by the formula: perimeter of the sector = 2 r + s, where r is radius and s is arc length of the sector. Then, we will calculate the required area of the sector by the formula: area = $\dfrac{\theta }{{{{360}^ \circ }}}\pi {r^2}$where $\theta = \dfrac{s}{r}$.

Complete step-by-step answer:
We are given the perimeter of the sector of the circle as 20 cm.

The area of the circle is also given as 25$\pi $sq. cm.
We are required to find the area of the sector.
Let us first find the radius of the circle. We know that the area of the circle is given by the formula:
Area of the circle: $\pi r^2$
Comparing it with the given value of area of the circle, we get
$ \Rightarrow $$\pi r^2$ = 25$\pi $
$ \Rightarrow $$r^2$ = 25
$ \Rightarrow $r = 5
Now, the perimeter of the sector is given by: 2 r + s, where r is the radius of the circle and s is the arc length of the sector. The perimeter of the sector is 20 cm.
$ \Rightarrow $20 = 2 (5) + s
$ \Rightarrow $20 = 10 + s
$ \Rightarrow $s = 10 cm
For the area of the sector, we have the formula: $\dfrac{\theta }{{{{360}^ \circ }}}\pi {r^2}$
where, $\theta = \dfrac{s}{r}$$ \Rightarrow $$\theta $= $\dfrac{{10}}{5} = 2$
putting this value in the area of the sector, we get
$ \Rightarrow $area of the sector = $\dfrac{2}{{{{360}^ \circ }}}\pi \left( {{5^2}} \right)$
In radians, 360° can be written as 2$\pi $.
$ \Rightarrow $area of the sector = $\dfrac{2}{{2\pi }}\pi 25 = 25$
Therefore, the area of the sector is 25 sq. cm.

Note: In such questions, you may go wrong while calculating for the arc length using the perimeter because you are required to find the area of the circle before that. Be careful while converting the degrees into the radian form as we did for the sake of simplicity in the answer.