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Im trying to figure this out. Like you could go one by one optimizing a square pyramid by minimizing its surface area for a given volume. Eventually when you get up to an decagon based right pyramid, it is evident that the pyramid is getting more and more optimal approaching the cone, hence the surface area is getting smaller and smaller.

However what is a faster way to do this? Perhaps making an nth based pyramid model and then differentiating it? I don't how that would exactly play out but its definitely possible.

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    $\begingroup$ This kind of problem is usually solved by the calculus of variations, not by differential calculus. But make sure your requirements are clear. Are you allowing any polygon at the base and any position for the apex, as long as the shape is a "pyramid" and the volume is as given? $\endgroup$
    – Rory Daulton
    Commented Jun 9, 2019 at 0:13
  • $\begingroup$ As the question states it is a right cone/ pyramid, so apex is at the center. $\endgroup$
    – Ahmed Anwer
    Commented Jun 9, 2019 at 2:24
  • $\begingroup$ the base must be a regular polygon, hence it being right cone/pyramid $\endgroup$
    – Ahmed Anwer
    Commented Jun 9, 2019 at 2:25

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