Two bodies, a sphere and a cube of same volume, which one has a larger surface area?
Surface area of a sphere and cube
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0The $i$dea 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. – 2011-06-06
2 Answers
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.
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.