Unfolding a Cube: Constructing an “Opened” Net via 3D Geometry

This example demonstrates how to construct and visualize an “opened” (unfolded) cube by systematically rotating and mirroring its faces around their shared edges. It transforms a 3D cube into a planar arrangement that preserves adjacency and hinge relationships between faces. Key concepts include solid geometry (cube faces and edges), angle constraints along common edges, and geometric transformations (rotations/mirrors) that map spatial relations to the unfolded configuration. Construction approach in GeoGebra: create points and the cube with Point and Cube; use Midpoint for reference geometry; employ Pyramid and Circle to support auxiliary structure and rotation placement; measure and specify hinge angles with Angle; perform the unfolding motions using Rotate; set up controlling references with Line and OrthogonalPlane; and generate corresponding mirrored face placements using Mirror. The construction tracks multiple named faces (e.g., faceABCD, faceADHE, faceABFE, faceBCGF, faceCDHG, faceEFGH) together with relevant edges (e.g., edgeAB, edgeAD, edgeBC, edgeCD, edgeCG, edgeGH) and uses a parameterized rotation angle (α) to control the opened state. Suitable for middle school to early university learners studying spatial geometry, cube nets, and transformation-based geometric modeling.