If a circle is elongated into an oval, with constant circumference, does the area remain constant?




The simple answer is no. I find it easier to fix the area and see what happens to the perimeter. Here is a 'thought experiment' that makes the trends clear.

You elongate it (make it an ellipse) by an affine transformation which shrinks the y coordinates and stretches the x coordinates by the same ratio. This preserves the area.

However, as you go to extreme ratios (1/100 and 100/1) you find that the perimeter is getting larger and larger, while the area is constant. In the limit, the perimeter is unbounded.

If you keep the perimeter constant, then the area will be getting smaller and smaller. In fact, the limit starting with a unit circle will be a double line of length \pi, and an area of 0.

Walter Whiteley