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Volume of a Rectangular Prism by Courtney

Volume of a Rectangular Prism by Courtney
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2017-04-17 12:25
PolygonSolid GeometryGeometric ConstructionSlider Animation
① This GeoGebra example dynamically displays a rectangular prism in 3D and computes its volume in real time, visually demonstrating the relationship between volume and the prism's length, width, and height. ② The core mathematical concept is the volume formula for a rectangular prism (V = length × width × height) in solid geometry, involving edges, faces, vertices, and volume measurement. ③ Construction highlights: from the construction summary, commands such as Line, OrthogonalLine, Circle, Intersect, Rotate, Angle, Segment, Midpoint, Polygon, Translate, Sequence, and Point are used to build the prism's framework, rotate the view, add auxiliary segments and labeled points, and form the solid surfaces via polygons poly1, poly2, poly3, with the volume stored in variable V. ④ Suitable for middle or high school solid geometry lessons, helping students grasp the geometric meaning of the volume formula and serving as an interactive tool for teachers to demonstrate volume calculation and spatial reasoning.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → a
  2. OrthogonalLine(A, a) → q_4
  3. Circle(A, 2) → r
  4. Intersect(r, q_4, 2) → E
  5. Intersect(r, a, 2) → B
  6. OrthogonalLine(E, q_4) → s
  7. OrthogonalLine(B, a) → t
  8. Intersect(s, t) → F
  9. Rotate(F, 45°, E) → S'
  10. Angle(F, E, S') → α
  11. Segment(E, S') → i_1
  12. Segment(E, F) → j_1
  13. Midpoint(j_1) → T
  14. Circle(E, T) → k_1
  15. Intersect(k_1, i_1, 1) → H
  16. Segment(E, H) → p_1
  17. Line(H, j_1) → q_1
  18. Line(F, p_1) → r_1
  19. Intersect(q_1, r_1) → V
  20. Segment(A, E) → s_1
  21. Segment(F, B) → t_1
  22. Line(B, r_1) → a_2
  23. Line(V, s_1) → b_2
  24. Intersect(a_2, b_2) → W
  25. Polygon(A, B, F, E) → poly1, p_2, r_2, s_2, q_2
  26. Polygon(B, W, V, F) → poly2, r_3, w, v, s_3
  27. Polygon(E, F, V, H) → poly3, q_3, s_4, v_1, u
  28. Translate(A, B - A) → A'
  29. Translate(E, B - A) → E'
  30. Sequence(Sequence[Polygon[Translate[A, Vector[(k * Vector[A, B]) + (l * Vector[A, E])]], Translate[B, Vector[(k * Vector[A, B]) + (l * Vector[A, E])]], Translate[F, Vector[(k * Vector[A, B]) + (l * Vector[A, E])]], Translate[E, Vector[(k * Vector[A, B]) + (l * Vector[E, A])]]], k, 0, o], l, 0, 0) → ligne
  31. Sequence(Sequence[Polygon[Translate[E, Vector[(l * Vector[A, B]) + (m * Vector[E, H])]], Translate[F, Vector[(l * Vector[A, B]) + (m * Vector[E, H])]], Translate[V, Vector[(l * Vector[A, B]) + (m * Vector[E, H])]], Translate[H, Vector[(l * Vector[A, B]) + (m * Vector[E, H])]]], l, 0, o], m, 0, q) → liste3
  32. Sequence(Polygon[Translate[B, (k * (W - B)) + (o * (B - A))], Translate[W, (k * (W - B)) + (o * (B - A))], Translate[V, (k * (W - B)) + (o * (B - A))], Translate[F, (k * (W - B)) + (o * (B - A))]], k, 0, q) → ligne2
  33. Sequence(Sequence[Polygon[Translate[A, Vector[(k * Vector[B, A]) - (l * Vector[E, A])]], Translate[B, Vector[(k * Vector[B, A]) - (l * Vector[E, A])]], Translate[F, Vector[(k * Vector[B, A]) - (l * Vector[E, A])]], Translate[E, Vector[(k * Vector[B, A]) - (l * Vector[E, A])]]], k, 0, 0], l, 0, p) → liste
  34. Sequence(Sequence[Polygon[Translate[B, Vector[((-l) * Vector[E, A]) + (m * Vector[E, H])]], Translate[W, Vector[((-l) * Vector[E, A]) + (m * Vector[E, H])]], Translate[V, Vector[((-l) * Vector[E, A]) + (m * Vector[E, H])]], Translate[F, Vector[((-l) * Vector[E, A]) + (m * Vector[E, H])]]], l, 0, p], m, 0, 0) → liste2
  35. Translate(E, (p * (E - A))) → E'_1
  36. Translate(F, (p * (E - A))) → F'
  37. Translate(V, (p * (E - A))) → V'
  38. Translate(H, (p * (E - A))) → H'
  39. Polygon(E'_1, F', V', H') → Haut, e'_1, f', v', h'
  40. Translate(A, ((o + 1) * (B - A))) → Sommet1
  41. Translate(A, ((p + 1) * (E - A))) → Sommet2
  42. OrthogonalLine(E'_1, s_1) → b
  43. OrthogonalLine(Sommet1, p_2) → c
  44. Intersect(b, c) → Sommet3
  45. Translate(A, ((q + 1) * (H - E))) → Sommet4
  46. OrthogonalLine(Sommet4, p_2) → d
  47. Segment(A, Sommet4) → e
  48. Segment(A, E'_1) → f
  49. Line(E'_1, e) → g
  50. Intersect(d, g) → C
  51. Line(Sommet3, g) → h
  52. Line(C, b) → i
  53. Intersect(h, i) → D
  54. Line(D, d) → j
  55. Line(Sommet1, h) → k
  56. Intersect(j, k) → G
  57. Segment(E'_1, Sommet3) → l
  58. Segment(Sommet3, Sommet1) → m
  59. Segment(Sommet3, D) → n
  60. Segment(D, G) → a_1
  61. Segment(D, C) → b_1
  62. Segment(C, E'_1) → c_1
  63. Segment(F', B) → d_1
  64. Segment(V', V) → e_1
  65. Segment(H', V') → f_1
  66. Segment(A, Sommet1) → g_1
  67. Point(g_1) → I
  68. Segment(I, G) → h_1
  69. Segment(F, E) → l_1
  70. Segment(F, B) → m_1
  71. Line(E, g_1) → n_1
  72. Intersect(n_1, m) → J
  73. Segment(F, J) → c_2
  74. Segment(F', V') → d_2
  75. Segment(B, W) → e_2
  76. Line(J, h_1) → f_2
  77. Intersect(a_1, f_2) → K
  78. Point(h_1) → L
  79. Segment(L, K) → g_2
  80. Segment(J, K) → h_2
  81. Line(H, h_1) → i_2
  82. Line(K, l) → j_2
  83. Intersect(i_2, j_2) → M
  84. Segment(M, K) → k_2
  85. Intersect(i_2, e_1) → N
  86. Segment(N, M) → l_2
  87. Intersect(r_1, j_2) → O
  88. Segment(F, O) → m_2
  89. Sequence(Segment[Translate[F, ((-k) * (A - B))], Translate[O, ((-k) * (A - B))]], k, 0, o) → seg4
  90. Line(N, b_1) → n_2
  91. Intersect(n_2, h_2) → P
  92. Segment(N, P) → t_2
  93. Line(V, t_2) → a_3
  94. Intersect(a_3, h_2) → Q
  95. Line(S', a_3) → b_3
  96. Intersect(b_3, h_2) → R
  97. Segment(V, Q) → c_3
  98. Polygon(F, V, N, M, K, J) → poly4, f_3, v_2, n_3, m_3, k_3, j_3
  99. Polygon(V, F, F', V') → poly5, v_3, f_4, f'_1, v'_1
  100. Segment(N, M) → d_3
  101. Segment(M, K) → e_3
  102. Sequence(Segment[Translate[F, ((-k) * (A - E))], Translate[E, ((-k) * (A - E))]], k, 0, p) → seg_1
  103. Sequence(Segment[Translate[F, (k * (B - A))], Translate[B, (k * (B - A))]], k, 0, o) → seg_2
  104. Sequence(Segment[Translate[F, ((-l) * (E - H)) + (o * (B - A))], Translate[B, ((-l) * (E - H)) + (o * (B - A))]], l, 1, q) → seg10
  105. Sequence(Segment[Translate[N, (l * (H - E))], Translate[P, (l * (H - E))]], l, -2, q - 3) → seg5
  106. Sequence(Segment[Translate[B, ((-k) * (B - F))], Translate[W, ((-k) * (B - F))]], k, 1, p) → seg_3

AI Drawing Prompt Reference

Draw a rectangular prism with adjustable length, width, and height sliders. Start by creating a point A as one vertex of the prism, then create a line a through A and the x-axis. Next, create a circle r with radius 2 centered at A, and find its intersections with q_4 and a to get points B and E. Use rotations, angles, line segments, midpoints, and other operations to build the prism's framework, forming three base polygons poly1, poly2, and poly3. Finally, use translations and rotations to create the 3D structure of the prism and display its volume V.
The prompt above was compiled from the original drawing commands and is for reference only.