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I just want to ask if all shapes that have maximum area also have maximum perimeter and if all solids that have maximum volume also have maximum surface area. Why is this so? I just find it interesting that when it comes to shapes and solids, two parameters are simultaneously maximized by just forming a relationship involving two variables.

For example, the maximum area and perimeter of a rectangle with one held constant and the other held variable has the length and width equal. This relationship is the same regardless of which parameter is picked as constant and variable. What other applications of maxima and minima besides shapes and solids have this sort of event where both can be maximized or minimized using the same relationship?

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    In the plane, if you keep the perimeter of a closed curve constant and want to maximize its area, then you're going to obtain a circle. If you restrict yourself to closed non-intersecting polygons, then you're going to get regular polygons (since they are the closest to a circle). You might also want to check the isoperimetric problem https://en.wikipedia.org/wiki/Isoperimetric_inequality – Stefan Lafon Dec 22 '22 at 01:22
  • If we fix the area of a rectangle, the square with that area has the minimum perimeter, not the maximum. The perimeter can get as large as we want with very skinny and long rectangles, and has no maximum. – aschepler Dec 22 '22 at 02:00
  • These 2 comments are correct, and it answers to the standard question : maximize area with constraint on perimeter, or maximize volume with constraint on area. But the question is different : if you start with a specific shape, and you increase one dimension, yes, it will increase both area and perimeter. – Lourrran Dec 22 '22 at 08:40

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