Question

The central angles of two sectors of circles of radii 7cm and 21cm are respectively ${120^ \circ }$ and ${40^ \circ }$. Find the area of the two sectors as well as the lengths of the corresponding arcs. What do you observe?

Answer 1

Hint: To solve this question one should need to know the concept of radius and arc concerning the angle theta ($\theta $). The basic formulas are required to solve this question.
We must use these formulas
$Area(\operatorname{Sec} tor) = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi \times {r^2}$ --(1)
where $\theta $is in degrees and $r$=radius in given units.
Arc Length of Sector = $r \times \theta $ --(2)
where $\theta $is in radians and $r$=radius in given units.

Complete step-by-step answer:
Step1: Collect all the data or information given in the question. The given information is:
Radius of circle 1(${r_1}$) = 7cm
Radius of circle 2(${r_2}$) = 21cm
Central angles of circle 1(${\theta _1}$) = ${120^0}$
Central angles of circle 2(${\theta _2}$) = ${40^0}$
Step2: Now we will understand the demand of the question which is to find the areas of a sector as well as the length of arcs of two circles.
Step3: The area formula we will use from the hint section which is given by equation (1).
$ \Rightarrow Area(\operatorname{Sec} tor) = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi \times {r^2}$
Now we will find the area for the first circle,
\[ \Rightarrow {(Area)_1} = \dfrac{{{{120}^0}}}{{{{360}^0}}} \times \pi {(7)^2}\]
\[\therefore {(Area)_1} = \dfrac{{49\pi }}{3}c{m^2}\]
Similarly, we will find the area for the second circle,
\[ \Rightarrow {(Area)_2} = \dfrac{{{{40}^0}}}{{{{360}^0}}} \times \pi {(21)^2}\]
\[ \Rightarrow {(Area)_2}\]$ = \dfrac{1}{9} \times \pi (441)$
\[\therefore {(Area)_2} = 49\pi c{m^2}\]\[\]
Step4: Now we will find the length of the two arcs.
\[ \Rightarrow Angle = \dfrac{{arclength}}{{radius}}\]
Where angle used in the formula needs to be in radian but the given angle in question is in degrees so we will first try to convert that to radians by using equation (3).
Angle for the first circle,
       $ \Rightarrow {120^0} = \left( {\dfrac{\pi }{{180^\circ }} \times 120^\circ } \right)radians$
       $\therefore {\theta _1} = \left( {\dfrac{{2\pi }}{3}} \right)radians$
Angle for the second circle,
      $ \Rightarrow {40^0} = \left( {\dfrac{\pi }{{180^\circ }} \times 40^\circ } \right)radians$
  $\therefore {\theta _2} = \left( {\dfrac{{2\pi }}{9}} \right)radians$
 Arc length of circle 1$({S_1}) = \dfrac{{2\pi }}{3} \times radiu{s_1}$
         $ \Rightarrow $Arc length of circle 1$({S_1})$$ = \dfrac{{2\pi }}{3} \times 7$
        $\therefore $Arc length of circle 1$({S_1})$$ = \dfrac{{14\pi }}{3}cm$
Arc length of circle 2$({S_2}) = \dfrac{{2\pi }}{9} \times radiu{s_2}$
$ \Rightarrow $Arc length of circle 2$({S_2}) = \dfrac{{2\pi }}{9} \times 21$
$\therefore $Arc length of circle 2$({S_2})$$ = \dfrac{{14\pi }}{3}cm$
Final Answer: The arcs of circle 1 and circle 2 are \[\dfrac{{49\pi }}{3}c{m^2}\] and $49\pi c{m^2}$ respectively. Similarly, the value of arc lengths for circle 1 and circle 2 is both $ = \dfrac{{14\pi }}{3}cm$.

Note: This question is pretty simple and based on the concept of circular arcs. So to solve it successfully students should be very clear about the basic numerical formulas and related terminologies of arc, such as arc length, area of arc, etc. Apart from this students should know about the conversion of angle from degrees to radians.
Formula to convert in degrees to radians and vice-versa.
$\pi (radians) = {180^0}(\deg rees)$