Physics

Three coins of the same size (radius $$1cm$$) are placed on table such that each of them touches the other two. The area enclosed by the coins is:


ANSWER

$$\left( \sqrt { 3 } -\cfrac { \pi }{ 2 } \right) { cm }^{ 2 }$$


SOLUTION
Radius of each coin $$=1\ cm$$
With all the three centres an equilateral triangle of side 2 cm is formed.
Area enclosed by coind $$=$$ Area of equilateral triangle $$-3\times$$ Area of sector of angle $$60^{o}$$
                                         $$=\dfrac{\sqrt3}{2}(2)^2-3\times\dfrac{60}{360}\times\pi(1)^2$$
                                         $$=\dfrac{\sqrt3}{4}\times4-3\times\dfrac{1}{6}\times\pi$$
                                          $$=\left(\sqrt3-\dfrac{\pi}{2}\right)\ cm^2$$
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