Visualizing the Torus–Sphere Volume Ratio Relation

This example demonstrates how to construct and visualize a torus–sphere volume ratio relationship by using 3D geometric intersections and marked parameters in a single GeoGebra scene. The key mathematical ideas are solid geometry, intersection geometry (curves as the trace of intersecting surfaces), volume and ratio relationships, and symmetry/transform-based reasoning; it also incorporates angle parameters (e.g., theta and phi) to describe which regions are being compared, along with geometric-mean style quantities when forming the displayed relation. Construction workflow: use the command Surface to build the torus (and related surfaces), apply Intersect to obtain the intersection curves between the torus surface and the sphere/other reference geometry, create the reference sphere with Sphere, and define key points and auxiliary segments using Point, Midpoint, and Segment. The resulting objects (parameters like R, r, theta1, phi1 and multiple labeled values) support the visualization and annotation of the volume-ratio relation. Suitable for secondary-to-university geometry instruction, exploratory learning, and math visualization projects focused on deriving solid-volume relationships from geometric constructions.