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Parts of circles are drawn inside a square as shown in the picture below. Prove that the area of the blue region is half the area of the square.

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Let ‘a’ be the side of the square

Area of the blue part in the half portion of the square is equal to half of the area of the circle having diameter ‘a’.

Area of the blue part in the half portion of the square = \(\frac{1}{2}\times\pi(\frac{a}{2})^2=\frac{\pi a^2}{8}\)

We must subtract area of two circles having diameter a/2 from the half the area of the square to get the area of remaining blue shaded part.

Area of remaining blue shaded part.

Area of blue shaded part.

i.e., area of the blue part is equal to half the area of the square.

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