Spherical Cross-Section: Intersection Curve of a Plane and a Sphere

This GeoGebra example demonstrates the cross-section formed when a plane intersects a sphere. Using commands such as Sphere, Plane, and IntersectPath, it constructs a sphere and a cutting plane to illustrate a key property in solid geometry: the intersection of a plane with a sphere is always a circle, whose radius decreases as the perpendicular distance from the plane to the center of the sphere increases. The construction includes multiple geometric elements — the sphere, the cross-sectional circle, orthogonal auxiliary lines, polygons, and angle annotations — providing a rich visual representation of the spatial relationships involved. Interactive buttons and draggable points allow users to reposition the plane or adjust the sphere in real time, enabling dynamic exploration of how the shape and size of the intersection curve change under different configurations. When the plane passes through the sphere's center, the cross-section is a great circle with maximum radius; as the plane moves away from the center, the circle shrinks until it vanishes entirely once the plane no longer intersects the sphere. This applet is well suited for secondary and undergraduate geometry courses, supporting inquiry-based learning in solid and analytic geometry by letting students discover fundamental properties through direct manipulation rather than passive observation.