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I was told in my class that the ratio of the area of a circle to area of a square should be greater than the ratio of the volume of a sphere to volume of a cube. But, I am not able to show this.

For the area of a circle to area of a square, I have: $\pi R^2 / R^2 = \pi$

For the volume of a sphere to volume a cube, I have: $\frac{4 \pi R^3}{3 R^3}= 4/3\pi$

But, the latter is greater than the former.

Could someone point out my error?

Thanks!

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  • $\begingroup$ But what is given here? $\endgroup$ – Dr. Sonnhard Graubner Jan 12 at 19:27
  • $\begingroup$ The radius of the sphere ? $\endgroup$ – Dr. Sonnhard Graubner Jan 12 at 19:28
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    $\begingroup$ If the circle is inscribed in the square, then the area of the square is $(2r)^2=4r^2.$ Similarly, If the sphere is inscribed in the cube, then the volume of the cube is $(2r)^3=8r^8$. $\endgroup$ – irchans Jan 12 at 19:31
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    $\begingroup$ Your error lies in not considering which square and cube we are talking of. You should rephrase the question, for instance, like this: "the ratio of the area of a circle to the area of the circumscribed square should be greater than the ratio of the volume of a sphere to the volume of the circumscribed cube". $\endgroup$ – Aretino Jan 13 at 14:57

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