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Exsphere of the Prism

Exsphere of the Prism
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2019-11-11 20:24
CirclePolygonSolid GeometryGeometric Construction
① This example demonstrates how to construct the circumscribed sphere (exsphere) of a regular hexagonal prism, where the sphere passes through all vertices of the prism, visualizing the spatial relationship between a prism and its circumsphere. ② Core mathematical concepts include prisms, circumscribed spheres, determination of sphere center and radius, and the condition for vertices to be concyclic in 3D space. ③ Construction highlights: using GeoGebra commands Polygon and Prism to build the hexagonal prism, Segment and Intersect to locate key intersection points, Sphere to generate the circumsphere, and Circle to display cross-sections; key objects include base polygon poly1, prism faces (e.g., faceABHG), edges (e.g., edgeAG), and the sphere l. ④ Suitable for high school solid geometry or introductory university spatial geometry, helping students understand the existence and construction of circumspheres, and enabling dynamic exploration of how the circumsphere changes with prism dimensions.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 6) → poly1, f, F, g, h, i, j, k, C, D, E
  2. Prism(poly1, 2) → a, G, faceDEKJ, faceEFLK, faceAFLG, faceGHIJKL, edgeAG, edgeBH, edgeCI, edgeDJ, edgeEK, edgeFL, H, edgeGH, edgeHI, edgeIJ, edgeJK, edgeKL, edgeGL, I, J, K, L, faceABHG, faceBCIH, faceCDJI
  3. Segment(B, K) → l
  4. Segment(H, E) → m
  5. Intersect(l, m) → M
  6. Sphere(M, C) → b
  7. Circle(H, G, L) → c
  8. Circle(B, A, F) → d

AI Drawing Prompt Reference

Draw a regular hexagonal prism and its circumscribed sphere. Start by creating a hexagonal base, then extend it vertically to form the prism. Identify a line segment from a base vertex to the corresponding vertex on the top face, and another from the same base vertex to a non-adjacent top vertex. Find the intersection of these two lines. Use this intersection point as the center to draw a sphere passing through all the prism's vertices, ensuring it touches the base center. Finally, add circles to the base and top faces to visualize the sphere's cross-sections.
The prompt above was compiled from the original drawing commands and is for reference only.