Tag: spaces and boundaries - 1

Questions Related to spaces and boundaries - 1

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:

maths spaces and boundaries - 1 area of rectangular paths comparing areas spaces and boundaries - 2

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

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

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

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

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Correct Option: B
Explanation:

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 $60^{o}$
                                         $=\dfrac{\sqrt3}{2}(2)^2-3\times\dfrac{60}{360}\times\pi(1)^2$
                                         $=\dfrac{\sqrt3}{4}\times4-3\times\dfrac{1}{6}\times\pi$
                                          $=\left(\sqrt3-\dfrac{\pi}{2}\right)\ cm^2$