Multiple choice

A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

  1. 2 : 3

  2. 3 : 4

  3. 1 : 4

  4. 1 : 2
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D Correct answer
Explanation

Let square side = a. Inscribed circle radius = a/2. Area = pi*(a/2)^2 = pi*a^2/4. Circumscribed circle radius = diagonal/2 = (a*sqrt(2))/2 = a/sqrt(2). Area = pi*(a/sqrt(2))^2 = pi*a^2/2. Ratio = (pi*a^2/4) / (pi*a^2/2) = 1/2.

AI explanation

Let the side of the square be s, meaning the inscribed circle has radius s divided by 2 and the circumscribed circle has radius s divided by the square root of 2. The area of the inscribed circle is pi times s squared divided by 4, and the area of the circumscribed circle is pi times s squared divided by 2. Their ratio is one half, making the final ratio 1 to 2.

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