question_answer

A regular hexagon is inscribed in a circle of radius 5 cm. If x is the area inside the circle but outside the regular hexagon, then which one of the following is correct?

A) \[13\,\,c{{m}^{2}}<x<15\,\,c{{m}^{2}}\]

B) \[15\,\,c{{m}^{2}}<x<17\,\,c{{m}^{2}}\]

C) \[17\,\,c{{m}^{2}}<x<19\,\,c{{m}^{2}}\]

D) \[19\,\,c{{m}^{2}}<x<21\,\,c{{m}^{2}}\]

Correct Answer: A

Solution :

OB = OA = radius

Also,     \[\angle AOB=60{}^\circ \]\[\left( \frac{360{}^\circ }{6}=60{}^\circ  \right)\]

and       \[\angle OAB=\angle OBA=60{}^\circ \]

\[\therefore \]\[\Delta AOB\]is an equilateral triangle.

Then,    \[AB=5\,\,cm\]

\[\therefore \]Area \[(x)=\]Area of circle \[-\]Area of hexagon

\[=\pi {{r}^{2}}-\frac{3\sqrt{3}{{(a)}^{2}}}{2}\]

\[=\frac{22}{7}\times {{(5)}^{2}}-\frac{3\sqrt{3}}{2}\times {{(5)}^{2}}[\because r=a=5]\]

\[=78.57-64.95=13.62\,\,c{{m}^{2}}\]