Four equal circles, each of radius a units, touch each other. Show that the area between them is sq units.

Here, first we join the centre of all adjacent circles then the distance between the centre of circles touching each other is equal to the side of the square formed by joining the centre of adjacent circles. Therefore, we can say that the side of the square equal to the twice of the radius of circle. 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 radius of each circle = “a” units

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

Side of square ABCD = 2×a units

where R = radius of circle

∴ Area all 4 quadrants = 4×Area of one quadrant

= πa^{2} sq. units → eqn2

Area of square = side×side = 2a×2a = 4a^{2}

⇒ Area of square = 4a^{2} sq. units → eqn3

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

⇒ Area of shaded region = 4a^{2} – πa^{2} (from eqn3 and eqn2)

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