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. 8 B. 4 C. 12 D. 10

GATE 2023 · General Aptitude · Circle · medium

Answer: Shaded area = 8 cm^2

  1. Identify all areas: Square area = 16 cm^2; side = 4 cm, radius of each arc = 2 cm = side/2
  2. Area of the top semi-circle (curved down): Top semi-circle area = 2*pi cm^2
  3. Area of the two bottom quarter-circles (curved up): Total area of both quarter-circle regions = 2*pi cm^2
  4. Compute the shaded area using complementary reasoning: The square is divided into 4 equal quarter-areas of 4 cm^2 each. The two bottom arcs together cover 2*pi cm^2 within the lower half (8 cm^2). The top semi-circle covers 2*pi cm^2 within the upper half (8 cm^2). The shaded region = upper half - top semi-circle that is outside + lower half covered by arcs = The shaded region = (top semi-circle area) + (lower square half - both arc areas) = 2*pi + (8 - 2*pi) = 8 cm^2
  5. Confirm the result: The pi terms cancel, giving an exact answer of 8 cm^2, independent of pi