Multiple choice

The maximum area of the rectangle that can be inscribed in a circle of radius $r ,$ is

  1. $\pi r ^ { 2 }$
  2. $r ^ { 2 }$
  3. $\pi r ^ { 2 } / 4$
  4. $2 r ^ { 2 }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A rectangle inscribed in a circle has diagonal equal to the diameter 2r. Let sides be x and y. x^2 + y^2 = (2r)^2 = 4r^2. Area A = xy. A^2 = x^2y^2. By AM-GM, (x^2+y^2)/2 >= sqrt(x^2y^2). 4r^2/2 >= xy. 2r^2 >= Area. Max area is 2r^2.

AI explanation

The maximum rectangle inscribed in a circle is a square whose diagonal equals the diameter of the circle, 2 * r. If the side of this square is s, then by the Pythagorean theorem s^2 + s^2 = (2 * r)^2, which gives 2 * s^2 = 4 * r^2. The area of the square is s^2, so s^2 = 2 * r^2, making the maximum area 2 * r^2.

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