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GATE 2023 CE (CE1) – Question 10

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

A square of side length 4 cm is given. The boundary of the shaded region is defined by one semi-circle on the top and two circular arcs at the bottom, each of radius 2 cm, as shown.
The area of the shaded region is _______ cm$^2$.

a 4 cm square. The shaded (white) region is bounded above by a semicircle of radius 2 cm centred at the middle of the square. At the bottom, two quarter-circle arcs of radius 2 cm centred at the two bottom corners cut off the black corner parts, and the arcs meet at the middle of the bottom edge.
  1. 8
  2. 4
  3. 12
  4. 10

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Show answer and explanation

Correct answer: (A) 8

Explanation

The black parts are the two top corner pieces and the two bottom quarter-discs. The top pieces together are the upper half of the square ($4 \times 2 = 8$) less the semicircle ($\frac{\pi \times 2^2}{2} = 2\pi$), that is $8 - 2\pi$. The two bottom quarter-discs have the area $2 \times \frac{\pi \times 2^2}{4} = 2\pi$. The black area is $8 - 2\pi + 2\pi = 8$, so the shaded area is $16 - 8 = 8$ cm².