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 |
\[=\pi {{r}^{2}}-\frac{3\sqrt{3}{{(a)}^{2}}}{2}\] |
\[=\frac{22}{7}\times {{(5)}^{2}}-\frac{3\sqrt{3}}{2}\times {{(5)}^{2}}[\because r=a=5]\] |
\[=78.57-64.95=13.62\,\,c{{m}^{2}}\] |
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