Question

The area of sector of a circle of radius 16cm cut off by an arc which is 18.5cm long is
(a) $168c{{m}^{2}}$
(b) $148c{{m}^{2}}$
(c) $154c{{m}^{2}}$
(d) \[176c{{m}^{2}}\]

Answer 1

Hint: Here, we are given a radius of circle with centre A as \[r=16cm\] and length of sector ADC as 18.5cm which is denoted by \[l=18.5cm\] . So, to find area of sector ADC we will use the formula \[Area=\dfrac{1}{2}\theta {{r}^{2}}\] and to find angle \[\theta \] we will use \[l=\theta r\] . Diagram of the question is a given below:

Complete step-by-step answer:
Now, according to the question, we are given all the necessary data i.e. radius \[r=16cm\] and length of sector ADC as 18.5cm which is denoted by \[l=18.5cm\] . So, for simplicity we will draw a figure which will get as,

Here, we have to find the area of the shaded part which will be obtained using the formula \[Area=\dfrac{1}{2}\theta {{r}^{2}}\] .
So, first we will find angle \[\theta \] , using the formula of length of arc given as \[l=\theta r\] .
\[\therefore l=\theta r\Rightarrow \theta =\dfrac{l}{r}\] ………………………….(1)
Now, using formula to find area of sector ADC as
\[Area=\dfrac{1}{2}\theta {{r}^{2}}\]
Substituting value of equation (1), we get
\[Area=\dfrac{1}{2}\times \dfrac{l}{r}\times {{r}^{2}}\]
\[Area=\dfrac{1}{2}\times l\times r\]
Now, putting the values \[r=16cm\] and \[l=18.5cm\] we get,
\[Area=\dfrac{1}{2}\times 18.5\times 16=18.5\times 8\]
\[Area=148c{{m}^{2}}\]
Thus, the area of sector of a circle of radius 16cm cut off by an arc which is 18.5cm long is \[148c{{m}^{2}}\] .
Hence, option (b) is the correct answer.

Note: Students sometimes do not understand that area of which sector i.e. minor sector or major sector is to be found out. So, instead of finding area of minor sector they find area of major sector by finding area of circle and from it subtracting area of minor sector.

Area of major sector \[=\pi {{r}^{2}}-\dfrac{1}{2}lr\Rightarrow r\left( \pi r-\dfrac{1}{2}l \right)\]
\[\Rightarrow 16\left( 3.14\times 16-\dfrac{1}{2}\times 18.5 \right)=16\left( 50.24-9.5 \right)\]
\[\Rightarrow 803.84-152=653.8c{{m}^{2}}\] . So, this answer is incorrect and also will not match with any given option. So, please be careful while solving this type of problem.