Multiple choice

On a square cardboard sheet of area $784$ $cm^2,$ four congruent circular plates of maximum size are placed such that each circular plate touches the other two plates and each side of the square sheet is tangent to two circular plates. Find the area of the square sheet not covered by the circular plates.

  1. $154$ $cm^{2}$
  2. $168$ $cm^{2}$
  3. $616$ $cm^{2}$
  4. $284$ $cm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Square side = sqrt(784) = 28. Four circles in 2x2 grid means each circle diameter = 14, radius = 7. Area of 4 circles = 4 * pi * 7^2 = 196 * pi. Using pi = 22/7, area = 196 * 22/7 = 28 * 22 = 616. Uncovered area = 784 - 616 = 168.

AI explanation

The side of the square is the square root of 784, which is 28 cm. Since four maximum congruent circles fit inside, their diameter is 28 divided by 2, making the radius of each plate 7 cm. The total area of the four circles is 4 times pi times 7 squared, giving 616 cm squared. Subtracting this from the square's area gives 784 minus 616, resulting in 168 cm squared of uncovered area.

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