areacirclesconic sectionsgeometry
For a given circle, is there exist an ellipse with same perimeter and area as to that circle?
If not, that is my suspicion, is in three-dimension parallel question: For a given sphere, is there exist an ellipsoid with same surface area and volume as to that sphere?
Actually, it is also true for squares (and for regular polygons in general!). The problem you ran into is what the equivalent of "r" is. The side length of a square is actually more comparable to the circle's diameter.
Instead, the correct analogue of the circle's radius is the distance from the center of the square to the midpoint of one side, which is only half as long as the square's side.
Here, we have $A = (2r)^2 = 4 r^2$ and $P = 4 (2r) = 8 r$. The perimeter is the derivative of the area with respect to $r$, just as in the case of a circle.
Place an electric charge at one focus of the ellipse and fixate it. Place a second charge of opposite polarity on the ellipse with frictionless support and impart the initial velocity that will cause its trajectory to trace the ellipse periodically. Measure the period, divide it by the desired number of regions, mark the positions of the second charge at intervals of the resulting time, and connect them to the focus. By Kepler's second law, the resulting areas will be equal.
Best Answer
No, because the isoperimetric quotient $A/P^2$ is smaller for a non-circular ellipse than for any circle.