Four equal circles are described about the four corners of a square so thateach touches two of the others. The shaded area enclosed between the circlesbeing 24/7 sq. cm, find the radius of the circles. [Use Π= ].

Question

Philoid · Accepted Answer

Let A, B, C and D be the four corners of the given square. Let the radius of each circle be 'a' cm such that the side of the square is '2a' cm

We know that,

Area of a sector , where 'r' is the radius of circle and θ is the angle of sector

For sector APS,

Angle of sector, θ = ∠ PAS = 90 °

[Each angle in a square is a right angle]

Radius of circle, r = 'a'

Putting the above values in formula, we get

Area of sector APS

By symmetry,

Area of sector APS is equal to area of each of sectors DRS, BPQ and CQR.

Also,

Area of shaded(Required) area = area of square ABCD - area of four sectors

⇒ 24 = 6a ²

⇒ a ² = 4

⇒ a = 2 cm

Hence, radius of each circle is 2 cm.