Question

Do 2 or more shapes with the same perimeter have to have the same area? Explain with examples. AND Do 2 or more shapes with the same area also have to have the same perimeter? Explain.

Answer 1

Yes. Take for example a square and an ellipsis with the same perimeter. The family of ellipses with the same perimeter can have any area between that of a circle to zero (if it is extremely “thin” i.e. if its eccentricity is large). The circle has the maximum area of any other shape with the same perimeter, so the square has the same area of one of the intermediate ellipses.