Question

Question 3

A circle of maximum possible size is cut from a square sheet of board. Subsequently, a square of maximum possible size is cut from the resultant circle. What will be the area of the final square?
(a) $\frac{3}{4}$ of original square (b) $\frac{1}{2}$ of original square
(c) $\frac{1}{4}$ of original square (d) $\frac{2}{3}$ of original square

Answer 1

Let a be the side of a square sheet.

Then, area of bigger square sheet $=a^2$ ...(i)
Now, we make the circle of maximum possible size from it.
Then, the radius of circle $\frac{a}{2}$ ...(ii)
So, its diameter (d) $=2.\frac{a}{2}=a$
Now, any square in a circle of maximum size will have the length of diagonal equal to the diameter of circle.
i.e. diagonal of square made inside the cirlce = a
So, the side of this square $=\frac{a}{\sqrt{2}}$ $[\because diagonal=side\sqrt{2}]$
$\therefore$ Area of this square $=\frac{a^2}{2}$ ...(iii)
From Eqs. (i) and (iii),
Area of final square is $\frac{1}{2}$ of original square.