What is the ratio of the area of a square inscribed in a circle to the square circumscribing the circle?

Guest Dec 28, 2014

#1**+5 **

Let the radius of the circle be r.

The diameter of the circle is then 2r, and this is the length of the diagonal of the inscribed square. This means the length of a side of the inscribed square is 2r/√2, so the area of the inscribed square is (2r/√2)^{2} = 2r^{2}.

The diameter is also the length of a side of the circumscribed square, so the area of the circumscribed square is (2r)^{2} = 4r^{2}.

Hence the ratio of the area of the inscribed square to that of the circumscribed square is 2r^{2}/4r^{2} = 1/2

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Alan
Dec 28, 2014

#1**+5 **

Best Answer

Let the radius of the circle be r.

The diameter of the circle is then 2r, and this is the length of the diagonal of the inscribed square. This means the length of a side of the inscribed square is 2r/√2, so the area of the inscribed square is (2r/√2)^{2} = 2r^{2}.

The diameter is also the length of a side of the circumscribed square, so the area of the circumscribed square is (2r)^{2} = 4r^{2}.

Hence the ratio of the area of the inscribed square to that of the circumscribed square is 2r^{2}/4r^{2} = 1/2

.

Alan
Dec 28, 2014