Multiple choice

The ratio of the area of a square inscribed in a semi-circle to that of the area of a square inscribed in the circle of the same radius is

  1. $2 : 1$
  2. $2 : 3$
  3. $2 : 5$
  4. $1 : 3$
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C Correct answer
Explanation

For a circle of radius R, the inscribed square has side s1 = R*sqrt(2), area = 2R^2. For a semi-circle of radius R, the inscribed square has side s2 = 2R*sqrt(5)/5, area = 4R^2/5. Ratio = (4R^2/5) / (2R^2) = 2/5.

AI explanation

Let the radius of the semicircle and circle be R. For the square inscribed in the semicircle, its diagonal forms a right triangle with the height R and half its side, so by the Pythagorean theorem, a^2 = R^2 + (a/2)^2, which gives a^2 = 4R^2 / 5. The area of this square is 4R^2 / 5. For the square inscribed in the full circle, its diagonal is 2R, so its area is half the square of the diagonal, which is (2R)^2 / 2 = 2R^2. The ratio of their areas is (4R^2 / 5) : 2R^2, which simplifies to 2:5. The ratio is 2:5.

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