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Four equal circles are described about the four corners of a square so that each touches two of the others, as shown in the figure. Find the area of the shaded region, if each side of the square measures 14 cm.

Here the distance between the center of circles touching each other is equal to the side of the square. Therefore, we can say that the radius of ach circle is equal to the half of the side of the square. Now by simply calculating the area of the 4 quadrants and then subtracting it from the area of the square we can easily calculate the area of the shaded region.

Given side of square = a = 14 cm

Central angle of each sector formed at corner = θ = 90°

So, radius of 4 equal circles = r = a/2 = 14/2

∴ Radius of 4 circles = r = 7 cm

where R = radius of circle

⇒ Area of all 4 quadrants = 49π → eqn2

Also, Area of square = side×side = a×a = a^{2} = 14^{2} (putting value of side of square)

⇒ Area of square = 196 cm^{2}→ eqn3

∴ Area of shaded region = Area of square – Area of all 4 quadrants

⇒ Area of shaded region = 196 – 49π (fromeqn3 and eqn2)

= 196 – (7×22)

= 196 – 154

= 42 cm^{2}

The area of shaded region is 42 cm^{2}.

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