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Volume Formula Derivation for Cuboids: Given Base Area and Height

Volume Formula Derivation for Cuboids: Given Base Area and Height
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2023-03-23 22:41
Solid GeometryGeometric ConstructionSlider Animation
In collections Quader und Würfel - Volumen
This example demonstrates calculating cuboid volume using base area G and height c. Core concepts include applying volume formula V=G×c and deriving it from unit cube decomposition. Construction uses Sequence commands to generate 3D cube grids (cubesHorizontal, cubesVertical), SpeechBubbleLeft/Right for student dialogues (Selina, Jule, Stef), and Textfield/input for interactive calculations. The extended problem explores finding cube volume from base area (9cm²). Suitable for grades 5-6 students learning volume concepts and formula application.

GeoGebra Commands & Parameters

  1. SpeechBubbleLeft(Pbubble1, 9, 1) → bubble1
  2. SpeechBubbleRight(Pbubble2, 6.7, 1.5) → bubble2
  3. Segment(A, B) → ab
  4. Segment(A, E) → ae
  5. Segment(B, F) → bf
  6. Segment(E, F) → ef
  7. Segment(A, D) → ad
  8. Segment(E, H) → eh
  9. Segment(D, H) → dh
  10. Segment(C, D) → cd
  11. Segment(B, C) → bc
  12. Segment(F, G) → fg
  13. Segment(C, G) → cg
  14. Segment(H, G) → hg
  15. SpeechBubbleLeft(Pbubble1 - (0, 0.1), 9.8, 1.1) → bubble3
  16. Textfield(input1) → InputBox1
  17. Sequence(Sequence[Polygon[A + (i, j), A + (i, j) + (cubeLengthWuerfel / 10, 0), A + (i, j) + (cubeLengthWuerfel / 10, cubeLengthWuerfel / 10), A + (i, j) + (0, cubeLengthWuerfel / 10)], i, 0, cubesHorizontal - 1], j, 0, cubesVertical - 1) → m1, {{0.9999999999999991, 0.9999999999999982, 1, 1}}
  18. Sequence(Sequence[Polygon[A + (i, cubesVertical) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)), A + (i, cubesVertical) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeLengthWuerfel / 10, 0), A + (i, cubesVertical) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeLengthWuerfel / 10 + cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10), A + (i, cubesVertical) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10)], i, 0, cubesHorizontal - 1], j, 0, cubesDepth - 1) → m2, {{0.3535533905932735, 0.35355339059327395, 0.35355339059327395, 0.35355339059327306}, {0.3535533905932744, 0.3535533905932722, 0.3535533905932766, 0.35355339059327306}, {0.35355339059327395, 0.35355339059327306, 0.35355339059327573, 0.3535533905932713}}
  19. Sequence(Sequence[Polygon[A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10 + cubeLengthWuerfel / 10), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (0, cubeLengthWuerfel / 10)], i, 0, cubesDepth - 1], j, 0, cubesVertical - 1) → m3, {{0.3535533905932766, 0.35355339059327306, 0.35355339059327484}}
  20. SpeechBubbleRight(Pbubble3, 11, 1.6) → bubble4
  21. Textfield(input2) → InputBox2
  22. Textfield(input3) → InputBox3
  23. SpeechBubbleLeft(Pbubble1 - (0, 2), 9.8, 3.1) → bubble5
  24. Textfield(input4) → InputBox4
  25. Textfield(input5) → InputBox5
  26. SpeechBubbleRight(Pbubble3, 6.3, 1.2) → bubble6
  27. SpeechBubbleRight(Pbubble3 - (0, 0.5), 11, 2) → bubble7
  28. SpeechBubbleRight(Pbubble3 + (0, 0.4), 11.5, 1.2) → bubble8
  29. Textfield(input6) → InputBox6
  30. Textfield(input7) → InputBox7
  31. Textfield(input8) → InputBox8
  32. Sequence(Sequence[Polygon[A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10 + cubeLengthWuerfel / 10), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (0, cubeLengthWuerfel / 10)], i, 0, cubesDepth - 1], j, 0, 1 - 1) → m9, {{0.3535533905932766, 0.35355339059327306, 0.35355339059327484}}
  33. SpeechBubbleRight(Pbubble3 + (0.2, -0.7), 11.5, 1.5) → bubble9
  34. SpeechBubbleLeft(Pbubble1, 12, 1) → bubble10
  35. Sequence(Sequence[Segment[A + (i, j), A + (i, j) + (cubeLengthWuerfel / 10, 0)], i, 0, cubesHorizontal - 1], j, 1, cubesVertical) → m11, {{0.9999999999999998, 1, 1, 1}}
  36. Sequence(Sequence[Segment[A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10)], i, 0, cubesDepth - 1], j, 1, cubesVertical) → m12, {{0.5000000000000003, 0.5000000000000003, 0.5000000000000003}}
  37. Polygon(A, E, F, B) → q1, a, e, f, b
  38. Sequence(Sequence[Polygon[A + (i, j), A + (i, j) + (cubeLengthWuerfel / 10, 0), A + (i, j) + (cubeLengthWuerfel / 10, cubeLengthWuerfel / 10), A + (i, j) + (0, cubeLengthWuerfel / 10)], i, 0, 4 - 1], j, 0, 1 - 1) → m7, {{0.9999999999999991, 0.9999999999999982, 1, 1}}
  39. Sequence(Sequence[Polygon[A + (i, 1) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)), A + (i, 1) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeLengthWuerfel / 10, 0), A + (i, 1) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeLengthWuerfel / 10 + cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10), A + (i, 1) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10)], i, 0, cubesHorizontal - 1], j, 0, 4) → m8, {{0.3535533905932735, 0.35355339059327395, 0.35355339059327395, 0.35355339059327306}, {0.3535533905932744, 0.3535533905932722, 0.3535533905932766, 0.35355339059327306}, {0.35355339059327395, 0.35355339059327306, 0.35355339059327573, 0.3535533905932713}, {0.35355339059327306, 0.35355339059327306, 0.35355339059327306, 0.35355339059327306}, {0.35355339059327395, 0.35355339059327484, 0.35355339059327395, 0.3535533905932722}}

AI Drawing Prompt Reference

Draw a cuboid to demonstrate the volume formula derivation. The cuboid is composed of multiple small cubes arranged in a grid, visually representing the volume as base area times height. Add speech bubbles around the cuboid to label the base area and height values, and show the calculation process and result. Include input fields below the cuboid for interactive calculation of volume by entering the base area and height.
The prompt above was compiled from the original drawing commands and is for reference only.