Open Middle: Maximizing Rectangular Prism Surface Area

This GeoGebra example demonstrates an interactive “open middle” exploration of a rectangular prism: under given constraints relating its side lengths, the model lets you vary the dimensions and visually track how the total surface area changes until it reaches a maximum. The key math ideas are solid geometry (rectangular prisms), surface area as a function of dimensions, constrained relationships among variables, and the optimization viewpoint (including how geometry supports reasoning about maxima). Construction highlights: Textfield is used to enter/drive parameters (e.g., length/width/height-type variables), Intersect determines constraint-consistent intersection points, Polygon builds the base and faces, and Prism generates the 3D solid. Segment defines essential edges, while Mode and If control conditional behavior for different construction states; Net supports auxiliary unfolding/representation where relevant. The example also labels and exposes faces and edges (e.g., faceADCF, faceAEGF, faceBEGH, faceBDCH, faceCFGH, edgeCD, edgeAF, edgeEG) and includes multiple intermediate variables (i1…i6) to link the geometry with the optimization/variation process. Suitable for high-school geometry classes, enrichment, and interactive learning of constrained maximization.