A circle maximizes area among 2D figures with a given perimeter due to its geometric properties, which can be derived through differential equations. Similarly, a sphere maximizes surface area among 3D figures with a fixed volume, following the same principles. The solutions to these problems involve the calculus of variations, which provides a framework for understanding these optimal shapes. Both cases illustrate how specific geometric configurations yield maximum area or surface area under defined constraints. Understanding these concepts is crucial for applications in mathematics and physics.
