February 18th, 2019, 11:13 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 458 Thanks: 29 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: 2,552 Thanks: 1402 
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: 458 Thanks: 29 Math Focus: Number theory 
Other than a square?

February 19th, 2019, 02:21 AM  #4 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 458 Thanks: 29 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: 645 Thanks: 408 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.

February 19th, 2019, 08:12 AM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 2,552 Thanks: 1402  

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