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Cylinder and Cone Volume Relationship Demo

Cylinder and Cone Volume Relationship Demo
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2021-11-05 22:59
FunctionTrigonometrySolid GeometryCalculusGeometric Construction
In collections Visual Geometry for Grade 9
This example dynamically demonstrates the volume relationship between a cylinder and a cone under the condition of equal base and height; core math concepts include volume formulas, geometric properties, and trigonometric function applications; construction highlights the use of GeoGebra commands/objects like Vector, Translate, OrthogonalLine, Segment, Mirror, Midpoint, Conic, Line, Intersect, Sequence, and Dilate; suitable for high school students to understand volume calculations and spatial reasoning skills development.

GeoGebra Commands & Parameters

  1. Vector(B_1, Y) → v
  2. Vector(B_1, X) → w
  3. Translate(O, Vector[(height * w)]) → C
  4. Vector(B_1, Z) → u
  5. Translate(O, Vector[(radius * u)]) → A_3
  6. OrthogonalLine(View3D, yAxis) → a_2
  7. OrthogonalLine(View3D, xAxis) → b_2
  8. Segment(O, A_3) → d
  9. Translate(O, Vector[(radius * v)]) → B
  10. Segment(O, B) → e
  11. Mirror(B, O) → D
  12. Mirror(A_3, O) → E
  13. Segment(D, O) → h_1
  14. Segment(E, O) → i
  15. Midpoint(B, A_3) → F
  16. Midpoint(B, F) → G
  17. Midpoint(O, G) → H
  18. Midpoint(H, G) → I
  19. Mirror(I, G) → J
  20. Conic(A_3, J, B, E, D) → f
  21. Segment(C, A_3) → j
  22. Segment(C, B) → l
  23. Segment(C, E) → m_2
  24. Segment(C, D) → n_2
  25. Line(F, O) → p
  26. Midpoint(B, E) → K
  27. Line(K, O) → q
  28. Intersect(f, p) → L, M
  29. Intersect(f, q) → N, P
  30. Segment(C, N) → r
  31. Segment(C, L) → s
  32. Segment(C, P) → t
  33. Segment(C, M) → a1
  34. Segment(C, O) → b1
  35. Vector(O, H1) → z
  36. Translate(f, z) → p1
  37. Translate(C, z) → I1
  38. Vector(H1, I1) → u1
  39. Translate(p1, u1) → q1
  40. Translate(D, z) → J1
  41. Translate(N, z) → K1
  42. Translate(A_3, z) → M1
  43. Translate(M, z) → N1
  44. Translate(L, z) → O1
  45. Translate(E, z) → P1
  46. Translate(P, z) → Q1
  47. Translate(B, z) → R1
  48. Translate(J1, u1) → S1
  49. Translate(K1, u1) → T1
  50. Translate(M1, u1) → U1
  51. Translate(N1, u1) → V1
  52. Translate(R1, u1) → W1
  53. Translate(Q1, u1) → Z1
  54. Translate(O1, u1) → A'2
  55. Translate(P1, u1) → B2
  56. Segment(S1, J1) → l1
  57. Segment(T1, K1) → m1
  58. Segment(U1, M1) → n1
  59. Segment(V1, N1) → r1
  60. Segment(W1, R1) → s1
  61. Segment(Z1, Q1) → t1
  62. Segment(B2, P1) → a2
  63. Segment(A'2, O1) → b2
  64. Segment(I1, H1) → g2
  65. Segment(H1, M1) → h2
  66. Sequence(Translate[p1, Vector[(i * u1)]], i, 0, 1, o) → L1, {-0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2923821738778212x + 4.0048495667262y = 3.927531160378909, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2900617318495289x + 4.130690419735488y = 4.0927178513682705, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2877412898212368x + 4.256531272744776y = 4.263015855446826, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2854208477929445x + 4.382372125754063y = 4.438425172614576, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2831004057646522x + 4.508212978763352y = 4.618945802871522, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2807799637363597x + 4.634053831772638y = 4.804577746217662, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2784595217080676x + 4.759894684781926y = 4.9953210026529975, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2761390796797754x + 4.885735537791214y = 5.191175572177527, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.273818637651483x + 5.011576390800502y = 5.392141454791252, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.271498195623191x + 5.137417243809789y = 5.5982186504941716, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2691777535948987x + 5.263258096819078y = 5.809407159286286, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2668573115666064x + 5.389098949828364y = 6.025706981167595, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2645368695383141x + 5.514939802837652y = 6.247118116138099, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.262216427510022x + 5.64078065584694y = 6.473640564197797, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2598959854817295x + 5.766621508856227y = 6.705274325346691, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2575755434534373x + 5.892462361865515y = 6.9420193995847805, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2552551014251452x + 6.018303214874803y = 7.183875786912063, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.252934659396853x + 6.144144067884091y = 7.430843487328542, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2506142173685606x + 6.269984920893378y = 7.6829225008342155, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2482937753402683x + 6.395825773902667y = 7.940112827429084, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2459733333119762x + 6.521666626911954y = 8.202414467113147, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.243652891283684x + 6.647507479921242y = 8.469827419886405, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2413324492553914x + 6.77334833293053y = 8.742351685748858, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2390120072270994x + 6.899189185939818y = 9.019987264700505, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.236691565198807x + 7.025030038949106y = 9.302734156741348, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2343711231705148x + 7.150870891958393y = 9.590592361871385, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2320506811422225x + 7.276711744967681y = 9.883561880090616, -0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2297302391139304x + 7.402552597976968y = 10.181642711399043}
  67. Sequence(Dilate[f, i, C], i, 0, 1, o) → L2, {?, -236.93085047769924x² + 43.782480955985015x y - 1195.2970838162364y² + 2151.7029247221644x + 5699.547347365541y = 12173.914664743255, -59.23271261942481x² + 10.945620238996254x y - 298.8242709540591y² + 538.373347312542x + 1400.6119809368158y = 2985.974576570776, -26.325650053077705x² + 4.864720106220559x y - 132.810787090693y² + 239.47598375313027x + 611.705388903221y = 1301.9401979333916, -14.808178154856202x² + 2.7364050597490635x y - 74.70606773851478y² + 134.8171448941359x + 338.0155672819193y = 718.4128181309682, -9.47723401910797x² + 1.7512992382394004x y - 47.81188335264945y² + 86.35459131336712x + 212.44598611569722y = 451.01312929472107, -6.581412513269424x² + 1.2161800265551395x y - 33.202696772673235y² + 60.01820131561615x + 144.8347285909487y = 307.211978006059, -4.8353234791367194x² + 0.8935200195098982x y - 24.39381803706605y² + 44.13154514061601x + 104.42756746236483y = 221.37783709481192, -3.7020445387140506x² + 0.6841012649372659x y - 18.676516934628694y² + 33.816190256534185x + 78.43517784433749y = 166.23401032267194, -2.925072228119744x² + 0.5405244562467286x y - 14.756754121188106y² + 26.741069641311007x + 60.774715536041015y = 128.81505107696034, -2.369308504776993x² + 0.43782480955985026x y - 11.952970838162367y² + 21.67817105474195x + 48.25652534801045y = 102.32641447394131, -1.9581062022950362x² + 0.3618386855866532x y - 9.87848829600196y² + 17.93064107439834x + 39.078951332089005y = 82.93259166224829, -1.6453531283173566x² + 0.3040450066387849x y - 8.300674193168312y² + 15.07915301757085x + 32.16287283030897y = 68.33767755866326, -1.4019576951343162x² + 0.25906793465079914x y - 7.072763809563535y² + 12.859103544723114x + 26.830498603232048y = 57.10051703751799, -1.2088308697841803x² + 0.22338000487747467x y - 6.0984545092665154y² + 11.096831446868414x + 22.639055307795598y = 48.28025012148535, -1.053026002123108x² + 0.19458880424882233x y - 5.312431483627718y² + 9.674530791618718x + 19.2895796298207y = 41.24200223048568, -0.9255111346785126x² + 0.17102531623431647x y - 4.6691292336571735y² + 8.50999958063365x + 16.5744374730132y = 35.545124271425735, -0.8198299324487863x² + 0.1514964738961419x y - 4.135976068568292y² + 7.544464903703177x + 14.345870482605887y = 30.876238611378078, -0.7312680570299359x² + 0.13513111406168213x y - 3.689188530297026y² + 6.735002536105621x + 12.496472672391434y = 27.0076973378676, -0.6563181453675877x² + 0.1212811106813989x y - 3.3110722543386037y² + 6.04967115298123x + 10.946697291513974y = 23.77101038733379, -0.5923271261942479x² + 0.10945620238996248x y - 2.98824270954059y² + 5.4643043768855675x + 9.636645746545671y = 21.039388489408648, -0.5372581643485241x² + 0.09928000216776643x y - 2.7104242263406713y² + 4.960345159370137x + 8.520541666666645y = 18.715999717181365, -0.48952655057375855x² + 0.09045967139666321x y - 2.4696220740004873y² + 4.52335264423602x + 7.562932750788662y = 16.725901215636497, -0.447884405439885x² + 0.08276461428352552x y - 2.259540801164907y² + 4.1419530138719045x + 6.736030298229556y = 15.010390047328375, -0.4113382820793388x² + 0.07601125165969617x y - 2.0751685482920763y² + 3.8070895987261117x + 6.017813548863122y = 13.522979714032846, -0.3790893607643185x² + 0.07005196952957597x y - 1.9124753341059766y² + 3.5114785174307883x + 5.3906578888430055y = 12.226489806570692, -0.35048942378357856x² + 0.06476698366269969x y - 1.7681909523908812y² + 3.2492078963346844x + 4.840328042764205y = 11.090910911994108, -0.32500802534663803x² + 0.06005827291630314x y - 1.6396393467986774y² + 3.015438960837106x + 4.355229538121161y = 10.09181791150468}
  68. Midpoint(h2) → J_1
  69. Midpoint(d) → L_1
  70. Dilate(f, a, C) → f'_2
  71. Translate(p1, Vector[(a / 3 * u1)]) → p1'
  72. Translate(p1, Vector[(1 / 3 * u1)]) → p1'_1
  73. Translate(p1, Vector[(2 / 3 * u1)]) → p1'_2
  74. Sequence(Dilate[f, i, C], i, a, 1, o) → liste11, {?, -236.93085047769924x² + 43.782480955985015x y - 1195.2970838162364y² + 2151.7029247221644x + 5699.547347365541y = 12173.914664743255, -59.23271261942481x² + 10.945620238996254x y - 298.8242709540591y² + 538.373347312542x + 1400.6119809368158y = 2985.974576570776, -26.325650053077705x² + 4.864720106220559x y - 132.810787090693y² + 239.47598375313027x + 611.705388903221y = 1301.9401979333916, -14.808178154856202x² + 2.7364050597490635x y - 74.70606773851478y² + 134.8171448941359x + 338.0155672819193y = 718.4128181309682, -9.47723401910797x² + 1.7512992382394004x y - 47.81188335264945y² + 86.35459131336712x + 212.44598611569722y = 451.01312929472107, -6.581412513269424x² + 1.2161800265551395x y - 33.202696772673235y² + 60.01820131561615x + 144.8347285909487y = 307.211978006059, -4.8353234791367194x² + 0.8935200195098982x y - 24.39381803706605y² + 44.13154514061601x + 104.42756746236483y = 221.37783709481192, -3.7020445387140506x² + 0.6841012649372659x y - 18.676516934628694y² + 33.816190256534185x + 78.43517784433749y = 166.23401032267194, -2.925072228119744x² + 0.5405244562467286x y - 14.756754121188106y² + 26.741069641311007x + 60.774715536041015y = 128.81505107696034, -2.369308504776993x² + 0.43782480955985026x y - 11.952970838162367y² + 21.67817105474195x + 48.25652534801045y = 102.32641447394131, -1.9581062022950362x² + 0.3618386855866532x y - 9.87848829600196y² + 17.93064107439834x + 39.078951332089005y = 82.93259166224829, -1.6453531283173566x² + 0.3040450066387849x y - 8.300674193168312y² + 15.07915301757085x + 32.16287283030897y = 68.33767755866326, -1.4019576951343162x² + 0.25906793465079914x y - 7.072763809563535y² + 12.859103544723114x + 26.830498603232048y = 57.10051703751799, -1.2088308697841803x² + 0.22338000487747467x y - 6.0984545092665154y² + 11.096831446868414x + 22.639055307795598y = 48.28025012148535, -1.053026002123108x² + 0.19458880424882233x y - 5.312431483627718y² + 9.674530791618718x + 19.2895796298207y = 41.24200223048568, -0.9255111346785126x² + 0.17102531623431647x y - 4.6691292336571735y² + 8.50999958063365x + 16.5744374730132y = 35.545124271425735, -0.8198299324487863x² + 0.1514964738961419x y - 4.135976068568292y² + 7.544464903703177x + 14.345870482605887y = 30.876238611378078, -0.7312680570299359x² + 0.13513111406168213x y - 3.689188530297026y² + 6.735002536105621x + 12.496472672391434y = 27.0076973378676, -0.6563181453675877x² + 0.1212811106813989x y - 3.3110722543386037y² + 6.04967115298123x + 10.946697291513974y = 23.77101038733379, -0.5923271261942479x² + 0.10945620238996248x y - 2.98824270954059y² + 5.4643043768855675x + 9.636645746545671y = 21.039388489408648, -0.5372581643485241x² + 0.09928000216776643x y - 2.7104242263406713y² + 4.960345159370137x + 8.520541666666645y = 18.715999717181365, -0.48952655057375855x² + 0.09045967139666321x y - 2.4696220740004873y² + 4.52335264423602x + 7.562932750788662y = 16.725901215636497, -0.447884405439885x² + 0.08276461428352552x y - 2.259540801164907y² + 4.1419530138719045x + 6.736030298229556y = 15.010390047328375, -0.4113382820793388x² + 0.07601125165969617x y - 2.0751685482920763y² + 3.8070895987261117x + 6.017813548863122y = 13.522979714032846, -0.3790893607643185x² + 0.07005196952957597x y - 1.9124753341059766y² + 3.5114785174307883x + 5.3906578888430055y = 12.226489806570692, -0.35048942378357856x² + 0.06476698366269969x y - 1.7681909523908812y² + 3.2492078963346844x + 4.840328042764205y = 11.090910911994108, -0.32500802534663803x² + 0.06005827291630314x y - 1.6396393467986774y² + 3.015438960837106x + 4.355229538121161y = 10.09181791150468}
  75. Sequence(Translate[p1, Vector[(i * u1)]], i, 0, a / 3, o) → liste1, {-0.3070623822190982x² + 0.056742095318956576x y - 1.5491050206258423y² + 1.2923821738778212x + 4.0048495667262y = 3.927531160378909}

AI Drawing Prompt Reference

Draw a cylinder and a cone with the same base radius and height. Start by creating a point O as the center of the base for both the cone and the cylinder, then create a point B as the apex of the cone. Next, create a cone using point O and point B, and create a cylinder using point O and a point A3 with the same radius as the cone's base. Then, create a vertical line segment from point O to point B, and another vertical line segment from point O to point A3, representing the heights of the cone and the cylinder. After that, create the side of the cone using point O and point B, and create the side of the cylinder using point O and point A3. Finally, create the base of the cone using point O and point B, and create the base of the cylinder using point O and point A3.
The prompt above was compiled from the original drawing commands and is for reference only.