Length of the minor arc of a circle of radius 10cm is 14cm. Find the area of the minor sector of the circle and length of the major arc?

Answer :

Given  : Length of minor arc  =  14 cm
And
Radius of the circle  =  10 cm
And
We know Arc length  = $\frac{Center angle}{180°}\times \mathrm{\pi r}$ , So

$$\begin{gathered} \Rightarrow \frac{\angle AOB}{180}\times \mathrm{\pi r} = 14 \\ \Rightarrow \frac{\angle \mathrm{A}\mathrm{O}\mathrm{B}}{180}\times \frac{22}{7}\times 10= 14 \\ \Rightarrow \frac{\angle \mathrm{A}\mathrm{O}\mathrm{B}}{18}\times \frac{22}{7}= 14 \\ \Rightarrow \frac{\angle \mathrm{A}\mathrm{O}\mathrm{B}}{9}\times \frac{11}{7}= 14 \\ \Rightarrow \angle \mathrm{A}\mathrm{O}\mathrm{B} = {\left(\frac{882}{11}\right)}^{°} \end{gathered}$$
And
We know area of sector of circle  = $\frac{Center angle}{360°}\times {\mathrm{\pi r}}^2$
So,
Area of minor sector = $\frac{{\left({\displaystyle \frac{882}{11}}\right)}^{°}}{360°}\times \frac{22}{7}\times 10\times 10 = \frac{882}{11 \times 360}\times \frac{22}{7}\times 10\times 10 = \frac{126 \times 2 \times 10 \times 10}{360} = \frac{126 \times 10}{18} = \mathbf{70}\mathbf{ }\mathbf{ }\mathbf{ }\mathit{c}{\mathit{m}}^{\mathbf{2}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathit{A}\mathit{n}\mathit{s}\mathbf{ }\mathbf{)}$
And
We know  circumference of circle  = $2\mathrm{\pi r}$
So,
Circumference of our circle  = $2\times \frac{22}{7}\times 10 = \frac{440}{7} = 62.857\approx 62.86 cm$
So,
Length of major arc  =  62.86  -  14  = 48.86 cm                                  ( Ans )