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To find the area of the largest square that can be fit in a circle, let’s see what happens when the square is drawn. We find that for a square all four corners of which lie on the circle, the diagonal of the square is equal to the diameter of the circle.
Here the radius is 6 cm and the diameter 12 cm. Now if a square has sides s, the diagonal has a length sqrt (s^2 + s^2) = s*sqrt 2
So s* sqrt 2 = 12
s = 12 / sqrt 2
The area of the square with side 12/ sqrt 2 is = 144 / 2 = 72.
Therefore the area of the largest square that can fit in a circle of radius 6 is 72.
We know that the equal chords are at equal distance from the centre. So if x is the chord length, then the perpendicular through the centre to the otherside is x. The maximum area of the square in the circle is a square with all 4 vertices on the circumferences.Therefore semi chord lengh and distance from the centre are equal in case of a square.
Therefore x^2+x^2 = r^2 .
So 2x^2 = r^2.
Or x = sqrt(r^2/2 ) = r/sqrt2, which is the unique side of the suare that could be inscribed in a circle having all its 4 vertices on the circumference. The area of the square is r^2/2 = 12^2/2 = 72 sq units, as the given radius is 12.
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