Consider the figure given below : PQRS is a square of side 1 unit and Q, S are the centres of the two circles. The area of the shaded portion is

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Q: 139 (IAS/1995)
Consider the figure given below : PQRS is a square of side 1 unit and Q, S are the centres of the two circles. The area of the shaded portion is

question_subject: 

Maths

question_exam: 

IAS

stats: 

0,2,9,4,1,4,2

keywords: 

{'circles': [0, 1, 0, 0], 'shaded portion': [0, 1, 0, 0], 'area': [0, 0, 0, 1], 'centres': [0, 0, 0, 1], 'square': [0, 0, 0, 1], 'figure': [0, 1, 1, 0]}

The area of the shaded region is found by subtracting the area of the semi-circles from the square. The area of the square PQRS with side 1 unit is 1x1 = 1 square unit. The total area of the two semicircles is half the area of a full circle with radius 0.5 units (since Q and S are the centers of the circles and QS is a side of the square). The area of such a circle is pi*(0.5)^2 = pi/4 square units. So, the area of the two semicircles combined is (pi/4)x2 = pi/2 square units.

So, if we subtract the area of the two semicircles from the area of the square, we get: 1 - pi/2 which clearly corresponds to option 4.

Dissolving the other options:

Option 1 is the area of the two semi-circles, but doesn`t consider the square and subtracting the overlapped area. Option 2 is just the half the area of the square. Option 3 isn`t taking two semicircles into account. Hence, none are correct.