question_answer

A circle is inscribed in a equilateral triangle of side a. The area of the circle is

A)  \[3\pi {{a}^{2}}\] sq unit    

B)  \[2{{a}^{2}}\] sq unit

C)  \[{{a}^{2}}\]sq unit      

D)  None of these

Correct Answer: A

Solution :

In an equilateral triangle length of median \[CD=\frac{\sqrt{3}}{2}a.\] Also, centroid of triangle lies on the centre of circle. \[\therefore \] \[OC=2\times \frac{\sqrt{3}}{2}a=\sqrt{3}a\] \[\therefore \] Area of circle =\[\pi {{(OC)}^{2}}\] \[=3\pi {{a}^{2}}\,\,sq\,\,unit\]