Webinars SET B - Grade 9-10 - Sunday@6-7pm EST-Recording Webinar 10

Video Transcription

Video Summary

The transcript covers a final class on 3D geometry and spatial reasoning through several contest-style problems.<br /><br />It begins with a warm-up on a sphere cut by three perpendicular great circles into 8 equal regions. A bee starts at an intersection and travels quarter-circle arcs, alternating left and right turns at each intersection. By tracking “left/right” on the sphere’s surface and using symmetry, the bee returns to its start after 6 quarter-circles.<br /><br />Next, the class reviews cube facts (6 faces, 8 vertices, 12 edges) and solves: cutting off all 8 corners adds 3 new edges per corner, so total edges become \(12+8\cdot3=36\).<br /><br />Then it discusses nets: identifying opposite faces and matching edges when folding cubes, rectangular prisms, tetrahedra, and octahedra. An octahedron-net question is solved by pairing obvious folds to find which labeled segment coincides with X.<br /><br />A “magic octahedron” problem uses equations plus the constraint that labels are the numbers 2–9 to deduce a value (notably finding \(A=7\)).<br /><br />The class then tackles painted-cube counting (maximizing cubes with exactly one painted face) and a reverse problem using 45 unpainted cubes and factorization to infer how many faces were painted.<br /><br />Finally, it shows how to convert 3D problems into 2D cross-sections (using perpendicular planes and special triangles), including hemisphere holes in a cube (diameter \(\sqrt2\)) and a folded net distance problem (answer \(1+\sqrt2\)).

Keywords

3D geometry

spatial reasoning

sphere great circles

cube corner cutting

polyhedron nets

octahedron net folding

magic octahedron

painted cube counting

3D to 2D cross-sections