Question

Find the area of the sector of a circle of radius 4 cm and angle \[{30^ \circ }\]. Find the area of the corresponding major sector. \[(\pi = 3.14)\]

Answer 1

Hint: Whether its area or circumference of a sector of the circle, we can get it by multiplying the ratio of the angle with the original formula say \[\theta \] is the angle of sector then the ratio will look like \[\dfrac{\theta }{{{{360}^ \circ }}}\].
Complete Step by Step solution:
As discussed in the hint we will get the formula for the area of sector.
Let's denote it by \[{A_s}\]
\[\therefore {A_s} = \pi {r^2} \times \dfrac{\theta }{{{{360}^ \circ }}}\]
Now it is given in the question that
\[\begin{array}{l}
radius = r = 4cm\\
\pi = 3.14\\
\theta = 30
\end{array}\]
Now put all of this in the formula \[{A_s} = \pi {r^2} \times \dfrac{\theta }{{{{360}^ \circ }}}\]
We will get it as
\[\begin{array}{*{20}{l}}
{{A_s} = \pi {r^2} \times \dfrac{\theta }{{{{360}^ \circ }}}}\\
{ \Rightarrow {A_s} = 3.14 \times {4^2} \times \dfrac{{{{30}^ \circ }}}{{{{360}^ \circ }}}}\\
{ \Rightarrow {A_s} = 3.14 \times 4 \times 4 \times \dfrac{1}{{12}}}\\
{ \Rightarrow {A_s} = \dfrac{{3.14 \times 4}}{3}}\\
{ \Rightarrow {A_s} = \dfrac{{12.56}}{3}}\\
{ \Rightarrow {A_s} = 4.186}
\end{array}\]
\[\therefore 4.186c{m^2}\] is the area of the sector we are told to find
Now for finding the sector of the area corresponding to the sector we just found we can do it by getting the difference between the total angle of the circle with the angle of sector we just dealt with i.e., \[{330^ \circ }\]
Now again everything remains the same just in place of \[\theta \] we will put \[{330^ \circ }\]instead of \[{30^ \circ }\]
\[\begin{array}{*{20}{l}}
{\therefore {A_{ms}} = 3.14 \times {4^2} \times \dfrac{{{{330}^ \circ }}}{{{{360}^ \circ }}}}\\
{ \Rightarrow {A_{ms}} = 3.14 \times 4 \times 4 \times \dfrac{{11}}{{12}}}\\
{ \Rightarrow {A_{ms}} = \dfrac{{3.14 \times 4 \times 11}}{3}}\\
{ \Rightarrow {A_{ms}} = \dfrac{{138.16}}{3}}\\
{ \Rightarrow {A_{ms}} = 46.053}
\end{array}\]
Where \[{A_{ms}}\] is the area of major sector

Note: The second corresponding area could be achieved in a different way also, all you need to is to find the area of the whole circle and then subtract the minor sector with angle \[{30^ \circ }\] which was already found as a result we will get the same output as we get now after rounding off to the nearest hundredths.