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

Answer: 3

$ABCDEF$ 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 $S=3$