Constant Perimeter, Changing Area? — Exploring Area Variation with Fixed Perimet

This GeoGebra example demonstrates a classic geometric question: can shapes with a constant perimeter have different areas? The answer is yes — for instance, a rectangle with a fixed perimeter can be stretched into a thin strip or compressed toward a square, yielding different areas, while a circle encloses the maximum area for a given perimeter. The construction includes polygons (poly1, poly2) and a circle, with draggable points or sliders that change the shape while the perimeter (perim) and area (areaT) are computed and displayed in real time. Core mathematical concepts include perimeter, area, extremum (isoperimetric inequality), and functional dependence between variables. Key GeoGebra commands used: Polygon, Area, Circle, Point, Segment, OrthogonalLine, Intersect, Perimeter, Distance, and If, along with auxiliary points (A, B, C, D, E, F, G, H, I, K), segments, perpendicular lines, intersection points, and text labels (text1). This is suitable for middle and high school students learning geometric measurement, function thinking, and basic optimization, as well as for intuitive exploration of the isoperimetric theorem.