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Section of a Tetrahedron Inside a Cube

Section of a Tetrahedron Inside a Cube
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2017-04-29 05:06
PolygonVectorCoordinate GeometrySolid GeometryGeometric Construction
①This example demonstrates how a tetrahedron is constructed inside a cube and how a cutting plane intersects it to reveal a dynamic cross-section. ②Core mathematical concepts include solid geometry, polyhedra, cross-sections, spatial relationships between a tetrahedron and a cube, and intersection points of a plane with edges. ③Construction details: The cube is created using the Cube command; vertices and intersection points are defined via Point and Intersect; the tetrahedron is built with Tetrahedron and Plane; the section polygon is generated using IntersectPath; Perimeter measures the section's boundary length; Vector and PointIn assist in positioning and analysis. ④Suitable for high school or early college students learning spatial geometry, and for teachers demonstrating interactive 3D constructions.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, zAxis) → A
  2. Point(xAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. Plane(A, F, C) → b
  6. Tetrahedron(F, C, b) → c, I, edgeFJ, edgeIJ, edgeCJ, J, faceCFI, faceFIJ, faceCFJ, faceCIJ, edgeFI, edgeCF, edgeCI
  7. IntersectPath(d, c) → poly1, K, L, M, N, f, g, h, i
  8. Perimeter(poly1) → perimeterpoly1
  9. PointIn(poly1) → Pointpoly1
  10. Vector(P, O) → u
  11. Vector(O, P) → v

AI Drawing Prompt Reference

Draw a cube in 3D space with one vertex at the origin and edges along the coordinate axes. Inside the cube, construct a tetrahedron whose vertices lie on the cube's vertices and faces. Use a cutting plane to intersect the tetrahedron, producing a dynamic cross-section polygon. Display the perimeter of this polygon. Place a point inside the section and draw vectors from this point to a fixed point in space to illustrate spatial relationships. Show the cube, tetrahedron, section, and vectors together in a clear 3D view.
The prompt above was compiled from the original drawing commands and is for reference only.