Question

On a square cardboard sheet of area $784$ $c{m}^2,$ 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 two circular plates. Find the area of the square sheet not covered by the circular plates.

• A. $154$ $c{m}^2$
• B. $168$ $c{m}^2$
• C. $616$ $c{m}^2$
• D. $284$ $c{m}^2$

Answer 1

The correct option is B $168$ $c{m}^2$

Let ABCD be the square sheet.

Let $a$ be the length of each side of the sheet.

Given area of sheet = 784 $c{m}^2$

$a^2=784$$c{m}^2$

$a=28$ cm

Diameter of each circular plate $=$ $\frac{a}{2}=\frac{28}{2}=14$ cm

Radius of each circular plate, $r=$ $\frac{14}{2}=7$ cm

Area of each circular plate $=\pi r^2=\frac{22}{7}\times 7^2=154$ $c{m}^2$

Therefore, area of $4$ circular plates $=4\times 154=616$ $c{m}^2$

Area of sheet not covered with circular plates $=$ Area of sheet $–$ Area of $4$ circular plates $=(784-616)c{m}^2=168$ $c{m}^2$