Question

On a square cardboard sheet of area $784c{m}^2$, four congruent circular plates of maximum size are placed such that each circular plate touches the other two plates and each side of the square sheet is tangent to the circular plates. Find the area of the square sheet not covered by the circular plates.

Answer 1

Let a be the side of Square $ABCD$,

Area of square $ABCD=784c{m}^2$

$\Rightarrow a^2=784$

$\Rightarrow a=\sqrt{784}$

$=\sqrt{2\times 2\times 2\times 2\times 7\times 7}$

$=2\times 2\times 7\Rightarrow a=28cm$

Refer image,

Now, in four circles,

$4r=AB$

$\Rightarrow 4r=28cm$

$\Rightarrow r=7cm$

Are enclosed between circles and squares

= Area of square - Area of $4$ circles

$=784-4\pi r^2$

$=784-4\times \frac{22}{7}\times 7\times 7$

$=784-616=168c{m}^2$

Hence, the area of square sheet not covered by cicle plates is $168c{m}^2$