The ratio of the area of a circle to the area of the largest square inscribed…

For a circle of radius r, the area is πr2. When a square is inscribed in the circle so that all four of its corners lie on the circle, the square's diagonal is exactly equal to the circle's diameter, 2r. This lets the square's side, and hence its area, be expressed purely in terms of r, so both areas can be compared using the same variable.

Working through the areas step by step:

1. Let the circle have radius r; its area is πr2.

2. Let the inscribed square have side a. Because all four corners touch the circle, the square's diagonal equals the circle's diameter, 2r.

3. Apply the Pythagorean theorem to the square's diagonal: a2 + a2 = (2r)2, so 2a2 = 4r2, giving a2 = 2r2.

4. The square's area is therefore a2 = 2r2.

5. Divide the circle's area by the square's area: πr2 / 2r2 = π/2 — the r2 terms cancel completely.

Checking with actual numbers confirms this: for r = 1, the circle's area is π ≈ 3.14 and the square's area is 2, giving a ratio of about 1.57. For r = 5, the circle's area is 25π ≈ 78.54 and the square's area is 50, giving the same ratio of about 1.57. Both cases produce the identical value π/2, verifying that the r2 terms cancel for any radius.

Since π/2 is a fixed number that never changes with the size of the circle, the ratio does not depend upon the circle chosen — it stays exactly π/2 whether the circle is small or large.