Question

A square carrom board with each side measuring 1 m is being painted. Calculate the area of the red section of the board.

• A. $[1-\pi (1-\frac{1}{\sqrt{2}}\left)\right]sq.m$
• B. $[1+\pi (1-\frac{1}{\sqrt{2}}\left)\right]sq.m$
• C. $[1-\pi (1+\frac{1}{\sqrt{2}}\left)\right]sq.m$
• D. $[1+\pi (1+\frac{1}{\sqrt{2}}\left)\right]sq.m$

Answer 1

The correct option is A $[1-\pi (1-\frac{1}{\sqrt{2}}\left)\right]sq.m$

The total area of the corners can be calculated as:


$Totalareaofthecorners=\pi \times (\frac{1}{2}{)}^{2}sq.m=\frac{\pi }{4}sq.m$

The area of the central circle can be calculated as:


$Areaofthecentralcircle=\pi (\frac{diameter}{2}{)}^2=\frac{\pi (d{)}^2}{4}=\frac{\pi }{4}\times (\sqrt{2}–\frac{1}{2}–\frac{1}{2}{)}^2=\frac{\pi }{4}\times (\sqrt{2}–1{)}^2=\frac{\pi }{4}\times (3–2\sqrt{2})$

The area of the red part can be calculated as:


$Area=1–\frac{\pi }{4}–\frac{\pi }{4}(3–2\sqrt{2})=1–\frac{\pi }{4}–\frac{3\pi }{4}+(–\frac{\pi }{4})\times (–2\sqrt{2})=1–\pi +\frac{\pi }{\sqrt{2}}=1–\pi (1–\frac{1}{\sqrt{2}})sq.m$