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Radius = r units
Area of the circle=πR²=πr² units² (not necessarily required)
Let the square be ABCD.
For it's area to be maximum ,
the length of diagonal needs to be equal to the diameter.
Diameter=2r units²=Diagonal AC or(BD)
Area=1/2 d²=2r² units².
Ans:The area of the square is 2r² units².

2 5 2

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If a square is inscribed in a circle, then the vertices of square are symmetrically placed on the circle. The diagonal of square becomes the diameter of the circle.

Let the side of square be "a".   Then √2 a = 2 r ,       a = √2 r

      Area of square = a² = 2 r²