question_answer

  The area of the circle circumscribing three circles of unit radius touching each other is:

A)  \[(\pi /3){{\left( 2+\sqrt{3} \right)}^{2}}\]

B)  \[6\pi {{\left( 2+\sqrt{3} \right)}^{2}}\]

C)  \[3\pi {{\left( 2+\sqrt{3} \right)}^{2}}\]            

D)  \[\left( \frac{\pi }{6} \right){{\left( 2+\sqrt{3} \right)}^{2}}\]

Correct Answer: A

Solution :

(a): Let A, B, C be centers of three circle having radius 1cm and O is the center of the circle circumscribing these three circles. \[AC=AB=BC=2cm\] By using the formula for circum-radius, we can calculate OC; \[OC=\frac{2}{3}\times \frac{\sqrt{3}}{2}\times 2=\frac{2}{\sqrt{3}}\] \[OX=OC+CX=\frac{2}{\sqrt{3}}+1\]