Largest Possible Area of a Rectangle with Fixed Perimeter

This GeoGebra example dynamically demonstrates the classic optimization problem: given a fixed perimeter, what rectangle dimensions yield the maximum area? It visually shows that a square achieves the largest area. Core mathematical concepts include quadratic functions, the arithmetic mean-geometric mean inequality, and the relationship between side lengths and area. The construction uses GeoGebra commands such as Segment, Point, Polygon, Intersect, and Vector to build the rectangle and control its dimensions, with text objects displaying live values of perimeter and area. Suitable for high school or introductory college students, this interactive tool helps learners observe how area changes as sides vary, reinforcing the algebraic proof with a geometric visualization. It can also serve as a pre-calculus introduction to optimization and derivatives.