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Cutting a Pyramid with an Interactive Plane

Cutting a Pyramid with an Interactive Plane
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2020-04-13 19:50
PolygonSolid GeometryTransformationGeometric ConstructionSlider Animation
In collections Cutting solids by a plane
This example demonstrates how an interactive draggable plane cuts a pyramid (constructed from cube vertices), showing in real time the resulting cross-section polygon and the volume of the remaining part. Core mathematical concepts include solid geometry intersections between planes and polyhedra, cross-section shapes (quadrilaterals), volume calculation, and geometric transformations (translation, reflection, scaling) as well as exponent relationships. Construction highlights: using GeoGebra commands such as Point, Cube, Midpoint, Line, Circle, Mirror, Plane, Center, OrthogonalLine, Vector, Translate, Segment, Intersect, Polygon, Sequence, Pyramid, and IntersectPath, the cube and its faces are built first, then a cutting plane is defined via points, lines, and planes, with IntersectPath generating the cross-section and volumes updating dynamically. Suitable for high school or early college students to explore spatial geometry properties of plane sections of polyhedra, and for teachers to demonstrate interactive geometric inquiry in class.

GeoGebra Commands & Parameters

  1. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  2. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, arêteAD, arêteAB, arêteBC, arêteCD, arêteAE, arêteDH, arêteEH, arêteEF, E, arêteBF, arêteFG, arêteCG, arêteGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  3. Midpoint(C, A) → I
  4. Midpoint(F, H) → J
  5. Line(J, I) → f
  6. Circle(I, Distance[A, B], f) → c
  7. Point(c) → K
  8. Midpoint(F, C) → M
  9. Mirror(K, I) → K'
  10. Line(K', K) → g
  11. Circle(I, (1.2 * Distance[A, B]), g) → d
  12. Point(d) → L
  13. Plane(L, K, K') → p
  14. Circle(K', L, K) → e
  15. Center(e) → N
  16. OrthogonalLine(N, p) → h
  17. Point(h) → O
  18. Vector(I, O) → u
  19. Translate(p, u) → p'
  20. Segment(K', K) → b
  21. Line(L, b) → f_1
  22. Point(arêteEF) → D_1
  23. OrthogonalLine(K, p) → h_1
  24. Plane(K', h_1) → p_1
  25. OrthogonalLine(K, p_1) → k_1
  26. Line(K', k_1) → l_1
  27. Intersect(f_1, l_1) → E_1
  28. Intersect(k_1, f_1) → F_1
  29. Mirror(F_1, K) → G_1
  30. Mirror(E_1, K') → H_1
  31. Polygon(F_1, E_1, H_1, G_1) → q2, m_1, e_1, n_1, q_1
  32. Translate(q2, u) → q2'
  33. Circle(I, Distance[I, A], f) → q
  34. Point(f) → S
  35. Segment(A, I) → i
  36. Segment(I, S) → j
  37. Sequence(Rotate[A, i, f], i, 0, (2 * π), (2 * π / Sides)) → l1
  38. Polygon(l1) → poly1
  39. Sequence(Segment[Element[l1, i], S], i, 1, Length[l1]) → l3
  40. Pyramid(poly1, S) → k, face1, face10, face11, face12, face13, face14, face15, face16, face17, face18, face19, face3, face20, face21, face22, face23, face24, face25, face26, face27, face28, face29, face4, face30, face31, face32, face33, face34, face35, face36, face37, face38, face39, face5, face40, face41, arête1, arête2, arête3, arête4, arête5, arête6, arête7, arête8, face6, arête9, arête10, arête11, arête12, arête13, arête14, arête15, arête16, arête17, arête18, face7, arête19, arête20, arête21, arête22, arête23, arête24, arête25, arête26, arête27, arête28, face2, arête29, arête30, arête31, arête32, arête33, arête34, arête35, arête36, arête37, arête38, face8, arête39, arête40, arête41, face9
  41. IntersectPath(p', k) → poly2, P, I_1, J_1, K_1, L_1, M_1, N_1, O_1, Q_1, R_1, S_1, Q, T_1, U_1, V_1, W_1, Z_1, A_2, B_2, C_2, D_2, E_2, R, F_2, G_2, H_2, I_2, J_2, K_2, L_2, M_2, N_2, O_2, T, P_2, Q_2, l, m, n, r, s, t, g_1, d_1, Z, i_1, j_1, r_1, s_1, t_1, a_1, b_1, c_1, f_2, g_2, A_1, h_2, i_2, j_2, k_2, l_2, m_2, n_2, p_2, q_2, r_2, B_1, s_2, t_2, a_2, b_2, c_2, d_2, e_2, f_3, g_3, h_3, P_1, i_3, j_3, k_3, C_1

AI Drawing Prompt Reference

Create an interactive demonstration of a plane cutting through a pyramid constructed from cube vertices, showing the resulting cross-section polygon and the volume of the remaining part in real time. Build the cube first, then define the cutting plane by dragging points. Use path intersection to generate the cross-section and dynamically update the volume. Suitable for exploring spatial geometry properties and transformations.
The prompt above was compiled from the original drawing commands and is for reference only.