Three equal circles each of radius …

In the given figure, three circles of radius 2 cm touch one another externally. These circle are circumscribed by a circle of radius R cm.We have to find the value of R and the area of the shaded region in terms of {tex}\pi{/tex} {tex}{/tex} and {tex}\sqrt 3{/tex}.

Clearly, {tex}\triangle{/tex}ABC is an equilateral triangle of side 4 cm.
In {tex}\triangle{/tex}BDO, we have,
{tex}\cos \angle O B D = \frac { B D } { O B }{/tex}
{tex}\Rightarrow \quad \cos 30 ^ { \circ } = \frac { 2 } { O B } \quad \left[ \because \angle O B D = 30 ^ { \circ } \right]{/tex}
{tex}\Rightarrow \quad \frac { \sqrt { 3 } } { 2 } = \frac { 2 } { O B }{/tex}
{tex}\Rightarrow \quad O B = \frac { 4 } { \sqrt { 3 } }{/tex}
{tex}\therefore \quad O P = O B + B P \Rightarrow R = \left( \frac { 4 } { \sqrt { 3 } } + 2 \right) \mathrm { cm }{/tex}
Let A be the area of the shaded region. Then,
A = Area of the larger circle of radius R - 3 x Area of a smaller circle of radius 2 cm + 3 (Area of a sector of angle 60° in a circle of radius 2 cm) - {Area of {tex}\triangle{/tex}ABC - 3 (Area of sector of angle 60° in a circle of radius 2 cm)}
{tex}\Rightarrow{/tex} A = Area of the larger circle of radius R - 3 x Area of a smaller circle of radius 2 cm + 6 x Area of a sector of angle 60° in a circle of radius 2 cm - Area of {tex}\triangle{/tex}ABC
{tex}\Rightarrow \quad A = \left\{ \pi \left( \frac { 4 } { \sqrt { 3 } } + 2 \right) ^ { 2 } - 3 \times π \times 2 ^ { 2 } + 6 \times \left( \frac { 60 } { 360 } \times \pi \times 2 ^ { 2 } \right) - \frac { \sqrt { 3 } } { 4 } \times 4 ^ { 2 } \right\} \mathrm { cm } ^ { 2 }{/tex}
{tex}\Rightarrow \quad A = \left\{ \pi \left( \frac { 16 } { 3 } + 4 + \frac { 16 } { \sqrt { 3 } } \right) - 12 \pi + 4 \pi - 4 \sqrt { 3 } \right\} \mathrm { cm } ^ { 2 }{/tex}
{tex}\Rightarrow \quad A = \left\{ \pi \left( \frac { 4 } { 3 } + \frac { 16 } { \sqrt { 3 } } \right) - 4 \sqrt { 3 } \right\} \mathrm { cm } ^ { 2 } = \left\{ \frac { 4 \pi } { 3 } ( 4 \sqrt { 3 } + 1 ) - 4 \sqrt { 3 } \right\} \mathrm { cm } ^ { 2 }{/tex}