Pyramid Cross Section with Inscribed Sphere

Pyramid Cross Section with Inscribed Sphere
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2015-07-14 13:45
① This GeoGebra example dynamically demonstrates the cross-section polygon of a pyramid (quadrilateral pyramid) cut by a plane, along with an inscribed sphere and related geometric elements. ② Core mathematical concepts include solid geometry cross-sections, inscribed spheres, vertices and edges of a pyramid, and measurements of distances and angles. ③ Construction highlights: uses Polygon and Pyramid commands to build the base and pyramid, Translate to create auxiliary planes, IntersectPath to obtain the intersection path, OrthogonalLine and Intersect for perpendiculars and intersections, Angle and Distance for measurements, Segment and PointIn for marking key segments and points, and text labels for clarity. ④ Suitable for high school solid geometry or college-level geometric visualization, helping students understand how the cross-section changes with the cutting plane and the relationship between the inscribed sphere and the pyramid.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Pyramid(poly1, E) → e, faceABE, faceBCE, faceCDE, faceADE, edgeAE, edgeBE, edgeCE, edgeDE
  3. Translate(xOyPlane, Vector[(3.751830268402151, -4.546624710067656, 3)]) → xOyPlane'
  4. IntersectPath(xOyPlane', e) → poly2, F, G, H, I, f, g, h, i
  5. OrthogonalLine(E, xOyPlane) → j
  6. Intersect(j, xOyPlane) → J
  7. Angle(j, xOyPlane) → α
  8. Intersect(j, xOyPlane') → K
  9. Distance(E, K) → distanceEK
  10. Segment(E, J) → k
  11. Distance(E, K) → l
  12. Distance(E, K) → m
  13. Distance(E, J) → distanceEJ
  14. PointIn(poly2) → Pointpoly2
  15. PointIn(poly1) → Pointpoly1
  16. Distance(A, B) → distanceAB

AI Drawing Prompt Reference

Draw a quadrilateral pyramid with an arbitrary quadrilateral base and an apex above it. Slice the pyramid with a plane parallel to the base to obtain a smaller, similar quadrilateral cross‑section. Drop a perpendicular from the apex to the base plane; mark its foot on the base and its intersection with the cutting plane. Show the distances from the apex to the cutting‑plane intersection, from the apex to the base foot, and the length of one base edge. Place a point inside the cross‑section polygon and another inside the base polygon.
The prompt above was compiled from the original drawing commands and is for reference only.