Great question.
To prove that the maximum area occurs when the shape is a square, we'd need to use some calculus (which is beyond the scope of the GRE).
So, let's just say it's a general property.

Cheers,
Brent

What are the possibilities for the inscribed rectangle?
The inscribed rectangle can be stretched and pulled to extremes: extremely
long and thin, extremely tall and narrow, and somewhere in between:
The “long and thin” and “tall and narrow” rectangles have a very small area,
and the “in between” rectangle has the largest possible area. In fact, the
largest possible rectangle inscribed inside a circle is a square
In this problem, the circumference is equal to 20π = 2πr. Thus r = 10. The
square then has a side length of 10\sqrt{2}and an area of (10\sqrt{2} )2 = 200.