On a square cardborad sheet of area $784 cm 2$ , 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.

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160 c m^{2}

168 c m^{2}

60 c m^{2}

None of these

answer is A.

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Detailed Solution

Given that, on a square cardboard sheet of area 784

{cm}^{2}

, four congruent 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.
We know that, the area of circle is

π r 2

.
Let ABCD be the square sheet, let be the length of each side of the sheet.
Finding the length of each side of the sheet,

a 2 = 784 c m 2

\Rightarrow a

= 28 cm
Finding the diameter and radius of the circular plate,

Diameter = a 2

⇒ 282

\Rightarrow

r = 142

\Rightarrow

7
Calculating the area of circular plate.

π r 2 = 227 × 7 × 7

\Rightarrow

154 cm

2

There are 4 congruent plates.
Calculating the area of sheet not covered with circular plates by subtracting the area of 4 circular plates from the area of sheet.
Area =