If three equal circles each of radius 1 cm are circumscribed by a larger circle, then the perimeter of circumscribing circle is

In the figure $\Delta ABC$ is an equilateral triangle

Given radius of smaller circle $=1\text{\hspace{0.17em}cm}$

In $\Delta ABD,\angle ABD=60°$

$\Rightarrow \angle OBD=\frac{60°}{2}=30°$

$\therefore$ in $\Delta BOD,$$\cos 30°=\frac{BD}{OB}$

$\Rightarrow \frac{\sqrt{3}}{2}=\frac{1}{OB}$

$\Rightarrow OB=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\text{ cm}$

$\therefore$ Radius of bigger circle $=\frac{2\sqrt{3}}{3}+1=\frac{2\sqrt{3}+3}{3}\text{ cm}$

$\therefore$ perimeter of bigger circle $=2\pi r$

$=2\pi \left(\frac{{\displaystyle 2\sqrt{3}+3}}{{\displaystyle 3}}\right)$

$=\frac{2\pi }{\sqrt{3}}(2+\sqrt{3})\text{ cm}$