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Using the Surface Area Formula for Cones

Using the Surface Area Formula for Cones
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2024-04-12 00:50
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates calculating the base area, lateral area, and surface area of right cones by unfolding and folding their nets. Key mathematical concepts include the geometric properties of cones and the surface area formula. GeoGebra commands used include Cone, Surface, and CurveCartesian. Suitable for middle school grades 6-8 to practice finding the surface area of cones.

GeoGebra Commands & Parameters

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.14149864841649745, 0.8908532169998348, 0.35539896478859767, 0.9097635199704961, 0.14484461822314743, 0.26268254082749337, 0.07681918249829224, 0.9273743433326141, 0.08221022048981301, 0.20834045061274428, 0.07793933518954511, 0.7170647996488183, 0.5038932747266234, 0.42406494535463746, 0.7625853014835271, 0.5680731254136597, 0.05352935888397227, 0.2921722176509615, 0.19191412831379062, 0.47981698601403294, 0.3255087940128001, 0.5756812499285179, 0.2516090957147377, 0.6848341128420116, 0.0751174641162054, 0.5513534870555367, 0.5260230924367495, 0.2889663514978131, 0.09217213512568936, 0.356340540238157, 0.6974584234392034, 0.840263834378866, 0.8636778166433376, 0.9879977106487265, 0.4821209059020314, 0.9147949287788881, 0.47047734950660425, 0.2329736359543627, 0.49732319154934773, 0.4514442232475814, 0.8869942502118968, 0.9204270456306648, 0.7072365765299317, 0.7788647020789672, 0.05198894403334271, 0.08514906020348433, 0.856903261764509, 0.9160219072254681, 0.5817533644284886, 0.009484747516006715, 0.8663199379828581, 0.32322656106810144, 0.855363071805623, 0.4582504459739406, 0.2981912132523375, 0.06319910574760353, 0.8844863634735787, 0.6483334591960946, 0.7744258883683527, 0.7715428693803135, 0.44781874270897815, 0.22433882669424102, 0.8276108448488892, 0.2030248993546215, 0.30246618375752105, 0.13351247640564817, 0.5145475068693462, 0.36427181007732756, 0.2815450281951064, 0.47722345368720825, 0.22947252500677692, 0.8048848438883849, 0.9388961375873951, 0.8825277669120491, 0.3648088928363273, 0.3075621314029103, 0.6881301056353707, 0.9757641968071605, 0.6372063390151194, 0.5435878349129116, 0.47422578727601244, 0.8807478458344489, 0.6162286639770382, 0.2530065953110786, 0.7415979322835754, 0.44692939962307765, 0.4223336621757881, 0.03894840867063198, 0.8129301808726844, 0.04787930653899176, 0.6197650244630586, 0.0029027742389083322, 0.6581388884017234, 0.4276323560956337, 0.048638528915957845, 0.8222629056064864, 0.0025880680581391426, 0.05179511689345907, 0.6088535829993539, 0.6231691961881323, 0.6881198624136651, 0.6918697711619598, 0.31649938964108276, 0.2705780294454572, 0.6095955445543958, 0.09498931067790622, 0.1911483669834381, 0.373565904022526, 0.6034601692728682, 0.29718608765527166, 0.37984247041073715, 0.317150249984265, 0.6633505444379638, 0.3564314716015135, 0.5206775152192956, 0.9123673184041466, 0.17095867902024472, 0.711986518271918, 0.19567729036106601, 0.47399430966805944, 0.8482444000488573, 0.26909439631175547, 0.5791242502102806, 0.8701361088477607, 0.2596588048808833, 0.8027226281326096, 0.4336757806146048, 0.39991958453190096, 0.4902963683286545, 0.840928473991531, 0.02244408879482966, 0.6647464122570863, 0.5178365729780542, 0.4374738598559957, 0.05075553747298879, 0.8649523751759906, 0.6727670261088744, 0.6745560287849385, 0.5065294010983666, 0.10730502228536236, 0.69269826811576, 0.009948204972837287, 0.8931282774534962, 0.12954287097837278, 0.39258647006327657, 0.3384594355621021, 0.6231320774112158, 0.2961186159691066, 0.04337326864016847, 0.2654258635395092, 0.22577446215335892, 0.8244549886073389, 0.6269333197343276, 0.2899358643317753, 0.6147553318290782, 0.7533559433966374, 0.43528807063418645, 0.12766876385571574, 0.846805035109514, 0.09473776165258216, 0.7368003339541687, 0.5675927957354877, 0.6147255719156115, 0.8265528919626524, 0.1415075939003723, 0.3733200718120654, 0.3816711870052049, 0.36867648981340706, 0.5918827644635981, 0.8186570817951412, 0.1661552968389907, 0.7646093435943647, 0.27412949707168144, 0.4578164511312695, 0.7996829955021346, 0.15012219906977423, 0.2686251316561352, 0.46387143618520543, 0.09100773933582232, 0.10542592090235874, 0.4763174984162672, 0.7943369901224415, 0.2427980404514483, 0.7100978499327998, 0.3107627159003966, 0.040251651902725234, 0.10797466389485373, 0.16982925297474857, 0.35305982060779695, 0.26481207991357136, 0.7841408628904735, 0.4510797493115187, 0.8391932797181301, 0.05947783314423351, 0.695377734094782, 0.8739340559218222, 0.5962384241041401, 0.7695155475674442, 0.6412141104158765, 0.44485246294711045}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. Shuffle(Sequence[5]) → possibleRads_{1}, {1, 5, 2, 3, 4}
  4. If(countQuestion_{1} ≟ 0, 2, Element[possibleRads_{1}, Mod[countQuestion_{1}, Length[possibleRads_{1}]] + 1]) → r1_{1}
  5. Shuffle(KeepIf[1 ≤ sqrt(CLIPBOARDmagicSTRINGkk^(2) - r1_{1}^(2)) ≤ 5, CLIPBOARDmagicSTRINGkk, Sequence[7]]) → possibleHsl_{1}, {3, 4, 5}
  6. If(countQuestion_{1} ≟ 0, 5, possibleHsl_{1}(1)) → Hsl_{1}
  7. Textfield(inputB) → InputBoxB
  8. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset
  9. Textfield(inputA) → InputBoxA
  10. Textfield(inputC) → InputBoxC
  11. Point(yAxis) → CA_{1}
  12. Point(yAxis) → CB_{1}
  13. Ray(A_{1}, (x(A_{1}), y(A_{1}), 40)) → h_{1}
  14. Point(yAxis) → B_{1}
  15. Point(Segment[(0, 0, 0.1), (0, 0, 4)]) → CD_{1}
  16. Random(3, 8) → nRand_{1}
  17. Circle(CA_{1}, CB_{1}) → d_{1}
  18. Segment((1, 0, 0), (1, 0, 5)) → aCon_{1}
  19. Circle(aCon_{1}, (0, 0, 0)) → cCon_{1}
  20. Intersect(xAxis, aCon_{1}) → KB_{1}
  21. Cone(KB_{1}, KA_{1}, 1) → bCon_{1}, dCon_{1}, eCon_{1}
  22. Intersect(yAxis, xAxis) → KC_{1}
  23. Segment(KC_{1}, KA_{1}) → lCon_{1}
  24. Distance(KB_{1}, KC_{1}) → rCon_{1}
  25. Angle(KA_{1}, KC_{1}, KB_{1}) → α_{1}
  26. Rotate(KA_{1}, (abs(tCil_{1}) * α_{1}), yAxis) → KAA_{1}
  27. OrthogonalLine(KAA_{1}, xOyPlane) → fCon_{1}
  28. Intersect(fCon_{1}, xOyPlane) → KBB_{1}
  29. Rotate(KC_{1}, bet_{1} / 2, fCon_{1}) → KCC_{1}
  30. Rotate(KC_{1}, ((-bet_{1})) / 2, fCon_{1}) → KCR_{1}
  31. CircumcircleArc(KCC_{1}, KC_{1}, KCR_{1}) → gCon_{1}
  32. Distance(KAA_{1}, KBB_{1}) → hCon_{1}
  33. Distance(KBB_{1}, KCC_{1}) → rcCon_{1}
  34. Surface(((kCon_{1} - ((v * rcCon_{1}) * cos((u * bet_{1} / 2)))) * cos(ϕCon_{1})) - (((1 - v) * hCon_{1}) * sin(ϕCon_{1})), ((v * rcCon_{1}) * sin((u * bet_{1} / 2))), (((1 - v) * hCon_{1}) * cos(ϕCon_{1})) + ((kCon_{1} - ((v * rcCon_{1}) * cos((u * bet_{1} / 2)))) * sin(ϕCon_{1})), u, -1, 1, v, 0, 1) → jCon_{1}
  35. CurveCartesian(((kCon_{1} - (rcCon_{1} * cos((u * bet_{1} / 2)))) * cos(ϕCon_{1})), (rcCon_{1} * sin((u * bet_{1} / 2))), ((kCon_{1} - (rcCon_{1} * cos((u * bet_{1} / 2)))) * sin(ϕCon_{1})), u, -1, 1) → mCon_{1}
  36. Segment(KE_{1}, KF_{1}) → nCon_{1}
  37. Segment(KD_{1}, KF_{1}) → pCon_{1}
  38. Corner(5) → SizeGV1_{1}
  39. Vector((1; (pi / 2 * tCil2_{1}) + (pi / 6 * (1 - tCil2_{1})); ((-pi)) / 2 + ((0.8 * pi / 2) * (1 - tCil2_{1})))) → view_{1}
  40. Rotate(jCon_{1}, -90°) → jCon'_{1}
  41. Rotate(KE_{1}, -90°) → KE'_{1}
  42. Rotate(KF_{1}, -90°) → KF'_{1}
  43. Segment(KE'_{1}, KF'_{1}) → nCon'_{1}
  44. Rotate(KD_{1}, -90°) → KD'_{1}
  45. Rotate(KF_{1}, -90°) → KF'_{2}
  46. Segment(KD'_{1}, KF'_{2}) → pCon'_{1}
  47. Rotate(cCon_{1}, -90°) → cCon'_{1}
  48. Rotate(mCon_{1}, -90°) → mCon'_{1}
  49. Rotate(dCon_{1}, -90°) → dCon'_{1}
  50. If(r1_{1} / Hsl_{1} ≤ 1 / 2, 1, 2) → factor_{1}
  51. Translate(KE'_{1}, Vector[(0, -1 + (2 * tCil2_{1}), 0)]) → EP1_{1}
  52. Translate(KF'_{1}, Vector[(0, -1 + (2 * tCil2_{1}), 0)]) → EP2_{1}
  53. Vector(Midpoint[EP1_{1}, EP2_{1}], EP1_{1}) → u_{1}
  54. Vector(Midpoint[EP1_{1}, EP2_{1}], EP2_{1}) → v_{1}
  55. Vector(Midpoint[EPr1_{1}, EPr2_{1}], EPr2_{1}) → w_{1}
  56. Vector(Midpoint[EPr1_{1}, EPr2_{1}], EPr1_{1}) → a_{1}
  57. Segment((0, -1, 0), (0, 0)) → f_{1}
  58. Segment(EP1_{1}, KE'_{1}) → g_{1}
  59. Segment(EP2_{1}, KF'_{1}) → i_{1}
  60. Segment(EPr2_{1}, (0, -1, 0)) → j_{1}
  61. Segment(EPr1_{1}, (0, 0, 0)) → k_{1}
  62. Text(r1_{1}, (0, 0), true, true) → labelRToMeasure_{1}
  63. Text(Hsl_{1}, (0, 0), true, true) → labelLToMeasure_{1}
  64. Vector(Corner[labelRToMeasure_{1}, 1], Corner[labelRToMeasure_{1}, 3]) → measureLabelR_{1}
  65. Vector(Corner[labelLToMeasure_{1}, 1], Corner[labelLToMeasure_{1}, 3]) → measureLabelL_{1}
  66. Text(Hsl_{1}, Midpoint[EP1_{1}, EP2_{1}] + (((-tCil2_{1}) * x(measureLabelL_{1}) / 2), (((2 * tCil2_{1}) - 1) * y(measureLabelL_{1})), (y(measureLabelL_{1}) * (1 - tCil2_{1}))), true, true) → labelL_{1}
  67. Text(r1_{1}, Midpoint[EPr1_{1}, EPr2_{1}] + ((-x(measureLabelR_{1})), y(measureLabelR_{1}) / 2), true, true) → labelR_{1}
  68. If(hhint_{1} ≤ hCon_{1}, tCil2_{1}^(2), sqrt(tCil2_{1})) → tCil_{2}
  69. Segment((1, 0, 0), (1, 0, 5)) → aCon_{2}
  70. Intersect(xAxis, aCon_{2}) → KB_{2}
  71. Intersect(yAxis, xAxis) → KC_{2}
  72. Segment(KC_{2}, KA_{2}) → lCon_{2}
  73. Distance(KB_{2}, KC_{2}) → rCon_{2}
  74. Angle(KA_{2}, KC_{2}, KB_{2}) → α_{2}
  75. Rotate(KA_{2}, (abs(tCil_{2}) * α_{2}), yAxis) → KAA_{2}
  76. OrthogonalLine(KAA_{2}, xOyPlane) → fCon_{2}
  77. Intersect(fCon_{2}, xOyPlane) → KBB_{2}
  78. Rotate(KC_{2}, bet_{2} / 2, fCon_{2}) → KCC_{2}
  79. Rotate(KC_{2}, ((-bet_{2})) / 2, fCon_{2}) → KCR_{2}
  80. Distance(KAA_{2}, KBB_{2}) → hCon_{2}
  81. Distance(KBB_{2}, KCC_{2}) → rcCon_{2}
  82. Rotate(Surface[((kCon_{2} - ((v * rcCon_{2}) * cos((u * bet_{2} / 2)))) * cos(ϕCon_{2})) - (((1 - v) * hCon_{2}) * sin(ϕCon_{2})), ((v * rcCon_{2}) * sin((u * bet_{2} / 2))), (((1 - v) * hCon_{2}) * cos(ϕCon_{2})) + ((kCon_{2} - ((v * rcCon_{2}) * cos((u * bet_{2} / 2)))) * sin(ϕCon_{2})), u, -1, 1, v, 0, 1], -90°) → jCon_{2}
  83. Segment(Rotate[KE_{2}, -90°], Rotate[KF_{2}, -90°]) → nCon_{2}
  84. Segment(Rotate[KD_{2}, -90°], Rotate[KF_{2}, -90°]) → pCon_{2}
  85. Rotate(CurveCartesian[((kCon_{2} - (rcCon_{2} * cos((u * bet_{2} / 2)))) * cos(ϕCon_{2})), (rcCon_{2} * sin((u * bet_{2} / 2))), ((kCon_{2} - (rcCon_{2} * cos((u * bet_{2} / 2)))) * sin(ϕCon_{2})), u, -1, 1], -90°) → basehint

AI Drawing Prompt Reference

Draw a right cone that can be unfolded and folded to show its base area, lateral area, and total surface area. The cone has a circular base and a vertex above the base, with a curved lateral surface. Include both the 3D cone and its net: a sector for the lateral surface and a circle for the base. The radius and height are randomly chosen. The unfolding is animated or controlled by a slider, with the cone rotating around its vertical axis.
The prompt above was compiled from the original drawing commands and is for reference only.