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Two bodies, a sphere and a cube of same volume, which one has a larger surface area?

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I removed the measure-theory tag since this is not really measure theory -- of course, the isoperimetric problem is one of the major motivations for the entire field of geometric measure theory, but I feel this question is a bit too basic to deserve to be called measure theory. By this I'm not implying at all that it isn't a good question, of course! – t.b. Jun 6 '11 at 8:42
The idea behind this problem is also kind of neat and intuitive; if you take a lump of clay (same volume for every figure) and make lots of different things out, then what kind of maneuvers on the clay increase/decrease surface area? I don't mean to actually ask this, but I thought it was kind of neat to think about. – james Jun 6 '11 at 10:21
up vote 5 down vote accepted

The sphere of radius $r$ has volume $\frac{4}{3}\pi r^3$ and surface area $4\pi r^2$ — the derivations of these formulas can be found on this Wikipedia page. The cube of the same volume as a sphere of radius $r$ has side-length $r \cdot \left(\frac{4}{3}\pi\right)^{1/3}$ and thus surface area $6 \cdot r^2 \cdot \left(\frac{4}{3}\pi\right)^{2/3}$. Since $ 6 \left(\frac{4}{3}\pi\right)^{2/3} \approx 15.6$ is bigger than $4 \pi \approx 12.6$ the answer is: The cube.

In fact, the sphere is the shape with minimal surface area among all bodies of the same volume, by the isoperimetric inequality.

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cube. Say they both have a volume of 1000cm3

First the cube Could measure 10cm by 10cm by 10cm So surface area = 6 x 100 = 600cm3 Now the sphere 1000cm3 = 4/3 π r3 r = 6.20350490899

Surface area = 4π r2 = 4 x π x 6.2030504908992 = 483.52673998cm3

So the cube has a higher surface area.

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