Question

Find the area of the sector of a circle of radius 5 cm, if the corresponding arc length is 3.5 cm.

Answer 1

Hint: First, we will use the formula of arc length $l = r.\theta $, where l is the arc length, r is radius of circle and $\theta $ is the angle of sector. Using this, we will have the value of angle which we have to put in $A = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi \times {r^2}$ to get the required value.

Complete step-by-step answer:
Let us draw a diagram for our reference. The diagram will be as follows:-

               

We have the arc length = l = 3.5 cm and r = 5 cm.
Now, we will use the formula of arc length which is $l = 2\pi r \times \dfrac{\theta }{{{{360}^ \circ }}}$, $\theta $where l is the arc length, r is radius of circle and is the angle of sector.
So, putting the values in the formula, we will get the following expression:-
\[3.5 = 2 \times \dfrac{{22}}{7} \times 5 \times \dfrac{\theta }{{{{360}^ \circ }}}\]
Switching LHS to RHS and vice versa, we will have:-
$\theta = {\left( {\dfrac{{3.5 \times 7 \times 360}}{{2 \times 5 \times 22}}} \right)^ \circ }$
On simplifying it, we will have:-
$\theta = {\left( {\dfrac{{441}}{{11}}} \right)^ \circ }$………………( 1)
Now, we also know the formula of area of a sector which is:
$A = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi \times {r^2}$, where A is the area, r is the radius and $\theta $ is the angle of sector.
Putting the value of r given in the question and the value from (1), we will have:-
$A = \dfrac{{{{\left( {\dfrac{{441}}{{11}}} \right)}^ \circ }}}{{{{360}^ \circ }}} \times \pi \times {5^2}$
Putting the value of $\pi = \dfrac{{22}}{7}$ and simplifying the expression, we will get:-
$A = \dfrac{{441}}{{11 \times 360}} \times \dfrac{{22}}{7} \times 25 = 8.75c{m^2}$.
Hence, the area is $8.75c{m^2}$.

Note: The students might making the mistake of angles here, because they might put the values in $A = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi \times {r^2}$ because in many books this formula is given. But you must remember that if we have angles in degrees then we have to use this formula and if we have angles in radians, we must use the formula mentioned in the solution and hint above. Do not think that there are so many formulas but this is just a way of changing the degree angle into radians by dividing it by 100.
The students may use the value of $\pi $ as per their comfort. Its exact value is $\dfrac{{22}}{7}$ which we used but we can also use the approximate value 3.14 depending upon the questions if not mentioned.