Multiple choice

$ABCD$ is a square with side $a$. With centres $A, B, C$ and $D$ four circles are drawn such that each circle touches externally two of the remaining three circles. Let $\delta$ be the area of the region in the interior of the square and exterior of the circles. Then the maximum value $\delta$ is:

  1. $a^2(1 - \pi)$
  2. $\displaystyle{a^2\left(\frac{4 - \pi}{4}\right)}$
  3. $a^2(\pi - 1)$
  4. $\displaystyle{\frac{\pi a^2}{4}}$
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B Correct answer
Explanation

The area of the square is a^2. Each circle has radius a/2. The area covered by the four sectors inside the square is 4 * (1/4 * pi * (a/2)^2) = pi * a^2 / 4. The area of the region in the interior of the square and exterior of the circles is a^2 - pi * a^2 / 4 = a^2(1 - pi/4) = a^2((4 - pi)/4).

AI explanation

Since the four circles of radius a/2 touch externally, the combined area of the four quarter circles inside the square equals the area of one full circle of radius a/2, which is pi * (a/2)^2 = pi * a^2 / 4. The area of the region exterior to the circles is the area of the square minus this circle area: a^2 - pi * a^2 / 4 = a^2 * (4 - pi) / 4.

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