An ellipse

1. Introduction The perimeter of an ellipse has always posed challenges due to the lack of a simple, closed formula. In this study, a new geometric method is proposed, based on an intuitive physical model that relates an ellipse to a square surrounded by an elastic circular ring under uniform pressure. --- 2. Practical Analogy Imagine an elastic circular ring surrounding a square with the same diameter as a circle. When uniform pressure is applied to the ring, it deforms into an ellipse, while the square inside it transforms into a rhombus. The lengths of its sides remain unchanged. The only change is in its angles and diameters, so that each pair of opposite angles are equal, and its diameters transform into a large and small axis. This is what happens. The more pressure is applied to the ring, the ring loses its perfect circular shape and conforms to an ellipse. The square cannot resist the pressure, so it shifts into a rhombus shape, while the length of each side remains constant. This analogy provides a tangible physical model that supports the mathematical derivation, fully explained with examples in the attached file. This derivation proves that the circumference of an ellipse erected on a rhombus with the same axes is equal to the circumference of a circle erected on a square with the same diameters and the same sides as the rhombus on which the ellipse is erected. Finally: This formula builds a bridge between simple geometric reasoning (physical deformation) and precise mathematical expression. This approach demonstrates how basic transformations under the action of uniform forces can lead to powerful and elegant mathematical relationships.