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GATE 2023 XH2 (English) – Question 10

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

A square with sides of length 6 cm is given. The boundary of the shaded region is defined by two semi-circles whose diameters are the sides of the square, as shown.

The area of the shaded region is _______ cm$^2$.

A square of side 6 cm. One semicircle has the left side as its diameter and the other has the bottom side as its diameter, both of radius 3 cm and bulging into the square. They overlap in a white lens near the bottom-left corner. The rest of the two semicircles is shaded.
  1. $6\pi$
  2. 18
  3. 20
  4. $9\pi$

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Show answer and explanation

Correct answer: (B) 18

Explanation

Each semicircle has area $\frac{\pi \cdot 3^2}{2} = 4.5\pi$. The two circles of radius 3 have centres $3\sqrt{2}$ apart, so their overlap (the white lens) is two circular segments of $90^\circ$ each: $2 \times \frac{9}{2}\left(\frac{\pi}{2} - 1\right) = 4.5\pi - 9$. The shaded area is the two semicircles without the lens: $2 \times 4.5\pi - 2(4.5\pi - 9) = 18$.