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If Saturn is 83x bigger that Earth by surface area why isn't the volume also 83x bigger than earth by volume? Is there a geometric or astronomical reason?

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    $\begingroup$ Surface area is proportional to the square of the radius. Volume, the cube of the radius. Saturn's radius is about 9 times Earth's. Each extra dimension requires the additional multiple of that 9 times the size. Surface area is 2 dimensions. Volume is 3 dimensions. $\endgroup$ – userLTK May 7 '17 at 18:43
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    $\begingroup$ Please put answers in answers, not in comments $\endgroup$ – James K May 7 '17 at 19:57
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    $\begingroup$ I'm voting to close this question as off-topic because it's very basic geometry and not at all specific to astronomy. $\endgroup$ – Warrick May 9 '17 at 13:39
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The surface area of a sphere is $A=4\pi r^2$ and the volume is $V=\frac 4 3 \pi r^3$.

Now the ratio $a = \frac {A_{saturn}} {A_{earth}} = (\frac {r_{saturn}} {r_{earth}})^2$

and the volume ratio is $b = \frac {V_{saturn}} {V_{earth}} = (\frac {r_{saturn}} {r_{earth}})^3$

The resulting ratio is $a = b ^{\frac 2 3}$.

User @Alex added that in your example, $764^{\frac 2 3}\approx83.6$, which I'd neglected to mention. Thank's @Alex.

And clearly that won't be equal to unless the two planets had the same radius.

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  • $\begingroup$ You could add that for this particular case $764^{2/3} = 83.6$... $\endgroup$ – Alex May 9 '17 at 13:29

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