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Pyramid and Cube Volume Relationship

Pyramid and Cube Volume Relationship
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2018-05-16 20:35
Solid GeometryGeometric Construction
This GeoGebra example demonstrates the volume relationship between a pyramid and a cube in 3D space. The core mathematical concept is solid geometry, specifically the formula that the volume of a pyramid is one-third of the volume of a cube with the same base and height. The construction uses GeoGebra commands such as Cube, Point, Line, Intersect, Pyramid, and Volume. Objects include a cube with faces (faceABFE, faceCDEF) and edges (edgeAB, edgeBF, etc.), and a pyramid with faces (faceHIJK, faceGHI, etc.) and vertices (G, H, I, J, K). The example is suitable for high school or early college students learning volume formulas, and can be used by teachers for classroom demonstrations or by students for interactive exploration. By adjusting the pyramid's apex, users can visually verify the constant volume ratio.

GeoGebra Commands & Parameters

  1. Point(Circle[(0.5, -0.5, 0), Distance[(-0.5, -0.5, 0), (0.5, -0.5, 0)], Segment[(-0.5, -0.5, 0), (0.5, -0.5, 0)]]) → A
  2. Cube((-0.5, -0.5, 0), (0.5, -0.5, 0), A) → a, B, faceABFE, faceCDEF, edge2, edge1, edge4, edgeAB, edge3, edgeBF, edgeCF, edgeCD, C, edge5, edgeDE, edgeAE, edgeEF, D, E, F, face1, face3, face2, face4
  3. Point(zAxis) → G
  4. Line(G, C) → f
  5. Line(G, D) → g
  6. Line(G, E) → h
  7. Line(G, F) → i
  8. Intersect(f, xOyPlane) → H
  9. Intersect(g, xOyPlane) → I
  10. Intersect(h, xOyPlane) → J
  11. Intersect(i, xOyPlane) → K
  12. Pyramid(H, I, J, K, G) → b, faceHIJK, edgeGH, edgeGI, edgeGJ, edgeGK, faceGHI, faceGIJ, faceGJK, faceGHK, edgeHI, edgeIJ, edgeJK, edgeHK
  13. Volume(b) → volumeb
  14. Point(b) → Pointb

AI Drawing Prompt Reference

Draw a cube in 3D space with its base on the horizontal plane)Skip. From a point above the cube, draw lines to the four base vertices of the cube, forming a pyramid whose base coincides with the cube's base. Place a movable point at the pyramid's apex and display the pyramid's volume. Use different colors for the cube's faces and the pyramid's faces to illustrate that the pyramid's volume is one-third of the cube's volume.
The prompt above was compiled from the original drawing commands and is for reference only.