Question

If the area and arc length of the sector of the circle is $60c{m^2}$ and $20cm$ respectively, then the diameter of the circle is
a) 6 cm
b) 12 cm
c) 24 cm
d) 36 cm

Answer 1

Hint: Divide the formula obtained for the arc length of the sector by the area of the sector formula to get the radius, from where the diameter of the circle can be calculated. The arc length of the sector is given by the equation \[L = \dfrac{\theta }{{{{360}^ \circ }}} \times 2 \times \pi \times r\]. The area of the sector is given by the equation $A = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi {r^2} $

Complete step-by-step solution:

From the given figure, the area noticeable as the slice of a pizza is called the sector of the circle. We have been given the values of the arc length and the area of the sector.
We know that the arc length of the sector is given by the formula, $L = \dfrac{\theta }{{{{360}^ \circ }}} \times 2 \times \pi \times r = 20cm ..........(1)$
where $L$ is denoted by the arc length, $\theta $ denotes the angle subtended by the arc and $r$ denotes the radius of the circle.
Now , similarly the area of the sector is given by the formula, $A = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi {r^2} = 60c{m^2} ..............(2)$
where $A$ is denoted by the area of the sector, $\theta $ denotes the angle subtended by the arc and $r$ denotes the radius of the circle.
Now dividing equation $(1)$ by $(2)$ , we get
$\dfrac{L}{A} = \dfrac{{20}}{{60}} = \dfrac{2}{r}$
So further simplifying the equation , we get $r = 6cm$
Hence we obtain the radius of the circle as 6cm.
The diameter of the circle is given by $d = 2r$
$d = 2 \times 6$ $ = 12cm.$
Hence the diameter of the circle is 12cm. Option B is the correct answer.

Note: In this problem, the main key is finding the ratio between the length of the arc and the area of the sector which will help us in obtaining the radius and consequently the diameter of the circle. Students usually go wrong with the formula to find the area of the sector and the arc length of the sector. Sometimes the student may even go wrong by finding the radius instead of diameter and choosing the wrong option.