Question

A regular hexagon is formed by two equilateral triangles inscribed in the circle $x^{2}+y^{2}=4$. If $S$ is the area of the hexagon (in sq. units), then find the greatest integer contained in $S$.

Answer 1

$ABCDEF$ is the regular hexagon formed by two equilateral triangles inscribed in the circle $x^{2}+y^{2}=4$.
$OM = OP \sin 30^{\circ} $
$\Rightarrow OM =2\left(\frac{1}{2}\right)=1$
$\frac{ OM }{ MF }=\tan 60^{\circ}$
$ \Rightarrow MF =\frac{1}{\sqrt{3}} $
$ \Rightarrow AF =\frac{2}{\sqrt{3}}$
Area of $\Delta OAF =\frac{1}{2}( OM )( AF )$
$=\frac{1}{2}(1)\left(\frac{2}{\sqrt{3}}\right)$
$=\frac{1}{\sqrt{3}}$
Area of hexagon $=6 \times$ Area of $\Delta OAF$
$S=6\left(\frac{1}{\sqrt{3}}\right)=2 \sqrt{3}$
$\Rightarrow$ Greatest integer contained in $S =3$