Question

If the area of a circle inscribed in an equilateral triangle is $4\pi c{m}^2$. then what is the area of the triangle?

• A. $12\sqrt{3}c{m}^2$
• B. $9\sqrt{3}c{m}^2$
• C. $8\sqrt{3}c{m}^2$
• D. $18c{m}^2$

Answer 1

The correct option is A $12\sqrt{3}c{m}^2$

Let $\Delta ABC$ be a equilateral triangle and a circle with centre O inscribed in it.
Area of circle $=4\pi c{m}^2$
$=>\pi r^2=4\pi$
$=>(OD{)}^2=4$
$=>OD=2$cm
OA is the angle bisector of $\angle A$ and $OD$ is the perpendicular bisector to side $AB$.
$\therefore \angle OAD={30}^0,\angle ADO={90}^0,\angle AOD={60}^0$
$\therefore \tan \angle AOD=\frac{AD}{OD}$
$=>\tan 60=\frac{AD}{2}$
$=>AD=2\sqrt{3}$
$=>AB=4\sqrt{3}$
Now, area of equilateral $△ABC=\frac{\sqrt{3}}{4}(AB{)}^2=\frac{\sqrt{3}}{4}\times 4\times 4\times 3$$=12\sqrt{3}c{m}^2$