Question

On a square cardboard sheet of area 784 cm², 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.

Answer 1

Let a be the side of square ABCD.

Area of square ABCD =784 cm²

⇒ `a^2 = 784`

⇒ `a = sqrt(784)`

= `sqrt(2 xx 2 xx 2 xx 2 xx 7 xx 7)`

= `2 xx 2 xx 7`

⇒ a = 28 cm

Now, in four circles,

4r = AB

⇒ 4r = 28 cm

⇒ r = 7 cm

Area enclosed between circles and square

= Area of square – Area of 4 circles

= `784 - 4pir^2`

= `784 - 4 xx 22/7 xx 7 xx 7`

= `784 - 616`

= 168 cm².

Hence, the area of square sheet not covered by circular plates is 168 cm².