Question

ABC is the equilateral triangle inscribed in a circle of radius 4 cm with center O. Find the area of shaded region

Answer 1

We have, $R=4cm$
$\therefore$ $AB=BC=CA=R$ $\sqrt{3}$ $=4$ $\sqrt{3}$ cm .............$[\because R=\frac{2}{3}handh=\frac{\sqrt{3}}{2}a\therefore R=\frac{a}{\sqrt{3}}]$

$\angle AOC=\angle ABC=2\times {60}^{\circ }={120}^{\circ }$

$\therefore$ Required area = $\frac{1}{3}$ (Area of the circle - Area of $\Delta$ ABC)

$\Rightarrow$ Required area = $\frac{1}{3}[\pi R^2-\frac{\sqrt{3}}{4}\times (sides{)}^2]$

$\Rightarrow$ Required area = $\frac{1}{3}[16\pi -\frac{\sqrt{3}}{4}\times (4\sqrt{3}{)}^2]$

$\Rightarrow$ Required area = $\frac{1}{3}(16\pi -12\sqrt{3})c{m}^2=\frac{4}{3}(4\pi -3\sqrt{3})c{m}^2$