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Cone, Cylinder & Sector with Equal Surface Area (2D/3D)

Cone, Cylinder & Sector with Equal Surface Area (2D/3D)
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2017-05-04 01:37
CircleAngleSolid GeometryGeometric ConstructionSlider Animation
This GeoGebra example demonstrates the geometric relationship where the surface areas of a cone, a cylinder, and a circular sector are equal, using linked 2D and 3D representations. The core mathematical concepts include lateral surface area formulas for cones and cylinders, sector area and arc length, and the unfolding of 3D solids into 2D nets. The construction employs GeoGebra commands such as Vector, Circle, Segment, Polygon, Cylinder, Angle, Cone, Circumference, Distance, PointIn, Plane, Rotate, CircleSector, and CircleArc, along with dynamic objects like sliders to adjust height and radius, enabling real-time comparison of computed areas. This resource is suitable for high school or early college students studying solid geometry, area derivations, and dynamic mathematical modeling, and serves as an excellent tool for classroom demonstrations or independent exploration of geometric relationships.

GeoGebra Commands & Parameters

  1. Vector(W_i, O) → diameter
  2. Vector(O, C_0) → r
  3. Circle(C_0, r_i, yAxis) → f
  4. Vector(H_i, O) → h_yAxisDown
  5. Vector(O, H_i) → h_YaxisUp
  6. Segment(O, H_i) → h_yAxis
  7. Vector(EdgeOfCylinder, W_i) → v_1
  8. Vector(W_i, EdgeOfCylinder) → h_edge
  9. Segment(H_i, EdgeOfCylinder) → d
  10. Vector(EdgeOfCylinder, H_i) → diameter_rev
  11. Vector(H_i, EdgeOfCylinder) → u_1
  12. Polygon(O, H_i, EdgeOfCylinder, W_i) → Two_dCylinderRep, TwoDCylinderRep_3, TwoDCylinderRep_2, TwoDCylinderRep_1, TwoDCylinderRep_4
  13. Segment(C_0, C_h) → TwoDCylinderRep_h
  14. Circle(C_h, r_i, yAxis) → g
  15. Cylinder((1 / 2 * w), (1 / 2 * w) + h, r_i) → Cylinder_1, c_i, c_h, CylinderSurface
  16. Vector(W_i, C_h) → s_1
  17. Angle(C_0, W_i, C_h) → b
  18. Vector(O, C_h) → s_2
  19. Segment(C_h, w) → s
  20. Angle(C_h, O, C_0) → b'
  21. Angle(O, C_h, w) → a
  22. Vector(C_0, C_h) → Height_cylinder
  23. Angle(C_h, C_0, O, xOyPlane) → c'
  24. Angle(C_h, C_0, W_i) → c
  25. Polygon(C_h, O, W_i) → Two_dConeFrameAsTriangle, Hypotenuse2_dMiniTriangleL, TwoDCylinderRep_0, o
  26. Cone(C_0, C_h, r_i) → Cone_i, c_iCone, ConeSurface
  27. Circumference(c_iCone) → pi_dCone
  28. Circle(circle_1, O, W_i) → circle_{2}Slice
  29. Circumference(circle_{2}Slice) → pi_dCircleEnd
  30. Distance(C_h, circle_1) → slant
  31. Segment(O, C_0) → radiusCone
  32. Circumference(circle_{2}Slice) → pi_dCircleEndSlice
  33. Circle(C_h, slant, xOyPlane) → t
  34. Circumference(t) → pi_dCircleT
  35. PointIn(t) → Pointt
  36. Segment(CutStart, C_h) → CompassLength
  37. Segment(C_h, CutStart) → cutFromLine
  38. Plane(circle_{2}Slice) → plane_Circle
  39. Rotate(CutStart, (-cutAngleMin), C_h) → CutFrom'
  40. Vector(C_h, CutStart) → v
  41. Vector(C_h, CutFrom') → v'
  42. CircleSector(C_h, CutStart, CutFrom', xOyPlane) → slice
  43. CircleArc(C_h, CutFrom', CutStart, xOyPlane) → ArcLengthDifferenceOfCircumferences
  44. Angle(CutFrom', C_h, CutStart) → θ
  45. CircleSector(C_h, CutFrom', CutStart, xOyPlane) → circle_2SliceInside
  46. Rotate(CutStart, (-ratio360deg), C_h) → CutDont
  47. CircleSector(C_h, CutDont, CutStart, xOyPlane) → e
  48. Angle(CutDont, C_h, CutStart, xOyPlane) → α