On a square cardboard 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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168

c m 2

192

c m 2

183

c m 2

142

c m 2

answer is A.

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Detailed Solution

Given that the area of the square cardboard is

784 cm 2

.
We have to find the area of the shaded region.

The expression to find the area of a circular geometry is

A = π r 2

, where r is the radius of a circle.
Side of the square cardboard, s is

784 = s 2 ∴ s = ± 28

As the length of the side of a square can’t be a negative value, we will consider only

s = 28 c m

.
Diameter and radius, r of the circular plate are

d i a m e t e r = 282 ∴ d i a m e t e r = 14 c m r = 142 ∴ r = 7 c m

Area of the four concurrent circular plates,

A s

A s = 4 × π r 2 ⇒ A s = 4 × 227 × 7 × 7 ⇒ A s = 616 c m 2

Therefore, the area of sheet not covered with circular plates, A’ is

A' = 784 − 616 ∴ A' = 168 c m 2

The area of sheet not covered with circular plates is 168

c m 2

.
Therefore, the correct option is 1.

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