Multiple choice

If the $4$ equal circles of radius $3 cm$ touch each other externally then the area of the region bounded by $4$ circles is ___ .

  1. $\displaystyle 4\left ( 9-\pi \right )cm^{2}$
  2. $\displaystyle 9\left ( 4-\pi \right )cm^{2}$
  3. $\displaystyle 5\left ( 6-\pi \right )cm^{2}$
  4. $\displaystyle 6\left ( 5-\pi \right )cm^{2}$
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B Correct answer
Explanation

Four circles of radius r=3 form a square of side 2r=6 connecting their centers. The area of the square is 6*6 = 36. The area of the four sectors inside the square is 4 * (90/360 * pi * r^2) = pi * r^2 = 9*pi. The region between them is 36 - 9*pi = 9(4-pi).

AI explanation

The centers of the four circles form a square with a side length equal to the sum of their radii, which is 3 + 3 = 6 cm. The area of this bounding square is 6^2 = 36 square cm. The combined area of the four quadrants inside the square equals the area of one full circle of radius 3 cm, which is pi * 3^2 = 9*pi. Subtracting the circular area from the square's area gives the bounded region as 36 - 9*pi, which factors to 9(4 - pi) square cm.

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