MathPlotAI Logo MathPlotAI

Net of a Cone APS - Dynamic 3D to 2D Unfolding

Net of a Cone APS - Dynamic 3D to 2D Unfolding
Loading the math board and drawing, please wait…
2019-05-24 02:30
CircleSolid GeometryTransformationGeometric ConstructionSlider Animation
① This GeoGebra example demonstrates the net of a cone, showing how the lateral surface unfolds into a sector and the base becomes a circle, linking 3D solid geometry to 2D planar representation. ② Core mathematical concepts include cone geometry, the relationship between the sector radius (slant height) and arc length (base circumference), and transformations such as rotation and angle measurement. ③ Construction highlights: commands like OrthogonalLine, Circle, Point, Intersect, Cone, Segment, Distance, Angle, Rotate, CircumcircleArc, Surface, and CurveCartesian are used to build the 3D cone and its unfolding; sliders (α, β, ϕ) control the rotation and unfolding animation, allowing interactive exploration. ④ Suitable for middle and high school geometry classes, introductory solid geometry, and math lab activities, helping students visualize the derivation of the cone's surface area formula and enhance spatial reasoning.

GeoGebra Commands & Parameters

  1. OrthogonalLine((1, 0, 0), xOyPlane) → a
  2. Circle(a, (0, 0, 0)) → c
  3. Point(a) → A
  4. Intersect(xAxis, a) → B
  5. Cone(B, A, 1) → b, d, e
  6. Intersect(yAxis, xAxis) → C
  7. Segment(C, A) → l
  8. Distance(B, C) → r
  9. Angle(A, C, B) → α
  10. Rotate(A, (t * α), yAxis) → A'
  11. OrthogonalLine(A', xOyPlane) → f
  12. Intersect(f, xOyPlane) → B'
  13. Rotate(C, β / 2, f) → C'
  14. Rotate(C, ((-β)) / 2, f) → C'_1
  15. CircumcircleArc(C', C, C'_1) → g
  16. Distance(A', B') → h
  17. Distance(B', C') → r'
  18. Surface(((k - ((v * r') * cos((u * β / 2)))) * cos(ϕ)) - (((1 - v) * h) * sin(ϕ)), ((v * r') * sin((u * β / 2))), (((1 - v) * h) * cos(ϕ)) + ((k - ((v * r') * cos((u * β / 2)))) * sin(ϕ)), u, -1, 1, v, 0, 1) → j
  19. CurveCartesian(((k - (r' * cos((u * β / 2)))) * cos(ϕ)), (r' * sin((u * β / 2))), ((k - (r' * cos((u * β / 2)))) * sin(ϕ)), u, -1, 1) → m
  20. Segment(E, F) → n
  21. Segment(D, F) → p

AI Drawing Prompt Reference

Create an animation showing the net of a cone: start with a 3D cone having a circular base and an apex above the base, displaying its height and slant height. Then unfold the lateral surface into a sector and the base into a circle, both lying in the same plane. The sector's radius equals the slant height, and its arc length equals the base circumference. Use sliders to control the unfolding and rotation, demonstrating the dynamic transition from solid to planar.
The prompt above was compiled from the original drawing commands and is for reference only.