Question

two concentric circles of radii 5cm and 8cm are shown below and sector form an angle of 60 degree at the centre O. What is the are of the shaded region.

Answer 1

Given:

two concentric circles of radii 5cm and 8cm are shown below and sector form an angle of 60 degree at the center O.

Solution:

Refer the question image,

Area of the shaded region = Area of the larger sector - Area of smaller sector,

Angle subtended by shaded region on the center is,

$\[ = 360^\circ - 60^\circ \]$

Know that,

Area of the sector$\[ = \pi {r^2} \times \frac{\theta }{{360^\circ }}\]$

Now calculate the area of the shaded region,

$\[ = \left\{ {\pi {R^2} \times \frac{{\left( {360^\circ - 60^\circ } \right)}}{{360^\circ }} - \pi {r^2} \times \frac{{\left( {360^\circ - 60^\circ } \right)}}{{360^\circ }}} \right\}\]$

$\[ = \pi \left[ {\left\{ {\frac{{300^\circ }}{{360^\circ }}\left( {{8^2} - {5^2}} \right)} \right\}} \right]\]$

$\[ = \pi \left( {64 - 25} \right)\left( {\frac{5}{6}} \right)\]$

$\[ = \pi \left( {39} \right)\left( {\frac{5}{6}} \right)\]$

$\[ \Rightarrow \frac{{195}}{6}\pi \,{\rm{c}}{{\rm{m}}^2}\]$

Hence, the correct answer is option (b).

The are of the shaded region $\[\frac{{195}}{6}\pi \,{\rm{c}}{{\rm{m}}^2}\]$.

Answer 2

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