# Prove that the area of a square inscribed in a circle is greater than the area of a rectangle inscribed in the same circle?

July 26, 2021 | education

| Since any rectangle inscribed in a circle has the following property: its diagonals are equal to the diameter of the circle d = 2 * r, and the area s = d * d / 2 * sin (<a), where <a is the angle between the diagonals.

For a simple rectangle:

s1 = d ^ 2/2 * (sin a), where sin a <1 for an arbitrary rectangle other than a square. (one)

For a square: s2 = d ^ 2/2 * sin a = d ^ 2/2 * sin 90 ° = d ^ 2/2 * 1 = d ^ 2/2. (2). Comparing formulas (1) and (2), we conclude that the area of a square is always greater than the area of a rectangle.

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