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Does the fact that a triangle has the maximum area for a given perimeter when it is an equilateral triangle (Isoperimetric Property of Equilateral Triangles) imply that the ratio of a triangle's area to its perimeter will have a maximum when all side lengths are equal? I found the derivative of Heron's formula for area divided be perimeter for an isosceles triangle and this is not the case. The ratio is at a maximum when the third side is $\sqrt5 - 1$ times the length of the two equal sides, which interestingly enough comes out to be $2(\phi - 1).$

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  • $\begingroup$ Will do, thank you. $\endgroup$ – John Singac Jul 11 '18 at 16:34
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    $\begingroup$ There's a problem of units when taking a ratio of area to perimeter. Depending on your choice of units the same triangle can have such a ratio that is arbitrarily large or arbitrarily small, and if you fix a unit of length, the ratio can be made arbitrarily large or arbitrarily small by changing the size of the triangle, $\endgroup$ – hardmath Jul 11 '18 at 16:37
  • $\begingroup$ Welcome to stackexchange. Are you sure your calculus/algebra is right? They symmetry of Heron's formula implies symmetry in the triangle for which the area is maximum. Perhaps the problem is in the units, as @hardmath comments. $\endgroup$ – Ethan Bolker Jul 11 '18 at 16:37
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Well, even for a square, your question is not very interesting. The maximum of area-perimeter ratio is $+\infty$. Just magnify your triangle or square, area increases quadratically, while perimeter only linearly increases.

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You should take a ratio of area to the perimeter squared to get meaningful (independent from the scale/units) results.

And yes, then the ratio is maximum for equilateral triangle.

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Area and perimeter have different unites so the ratio changes with the size of triangle.

The ratio of the area of an equilateral triangle of size $a$ to its perimeter is $$R=\frac {a\sqrt 3}{12}$$

As you see, the ration grows with the size of triangle.

For $a=4\sqrt 3$ we have $R=1$ and the ratio grows to $\infty$ as the triangle enlarges.

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