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 February 18th, 2019, 11:13 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory Rectangle inscribed in a circle Is the area of a rectangle inscribed in a circle ever rational?
 February 18th, 2019, 11:41 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,382 Thanks: 1281 put a $1 \times 1$ square inside a circle with radius $\dfrac{1}{\sqrt{2}}$ is one of infinite examples.
 February 19th, 2019, 02:17 AM #3 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory Other than a square?
 February 19th, 2019, 02:21 AM #4 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory Mea culpa. You're right, romsek.
 February 19th, 2019, 04:42 AM #5 Senior Member   Joined: Sep 2016 From: USA Posts: 578 Thanks: 345 Math Focus: Dynamical systems, analytic function theory, numerics Take any rectangle with rational area. Circumscribe it by a circle. This is always possible since rectangles are cyclic quadrilaterals. Thanks from Loren
February 19th, 2019, 08:12 AM   #6
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Joined: Sep 2015
From: USA

Posts: 2,382
Thanks: 1281

Quote:
 Originally Posted by Loren Other than a square?
SDK has answered this but it should be clear that because the distance from the center of a rectangle to each of the corners is a constant that any rectangle can be inscribed in a circle.

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