# Why does the cylinder with minimum surface area have a height equal to its diameter?

I'm trying to understand as to why, with a given volume, the diameter and height of a cylinder need to be the same for the minimum surface area. This thread, shows how to derive the minimum surface area of a cylinder with a given volume, but doesn't explain why this is.

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Volume $V=\pi r^2 h$.

Surface area $$S =2 \pi r h +2 \pi r^2$$

Now, the AM-GM inequality says

$$\frac{ \pi r h + \pi rh +2 \pi r^2}{3} \geq \sqrt[3]{\pi r h \cdot \pi rh \cdot2 \pi r^2}= \sqrt[3]{2 \pi V^2 }$$

with equality if and only if $\pi r h = \pi rh =2 \pi r^2$.

Basically, it boils down to this simple principle (which is more or less AM-GM): if some expressions have a constant product, their sums is minimal if the functions are equal (if they can be equal).

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See here artofproblemsolving.com/Wiki/index.php/Proofs_of_AM-GM for proof of the AM-GM inequality.The proof by Cauchy is the one I really like. –  Amitabh Udayiman May 6 '12 at 18:20