MathPlotAI Logo MathPlotAI

Net of a Cylinder by danielhenry (2)

Net of a Cylinder by danielhenry (2)
Loading the math board and drawing, please wait…
2020-02-22 06:04
CircleSolid GeometryTransformationGeometric ConstructionSlider Animation
① This GeoGebra example dynamically demonstrates the unfolding of a cylinder's surface into a flat net, showing how the lateral surface becomes a rectangle and the two bases become circles. ② Core mathematical concepts include solid geometry of cylinders, nets, surface area calculation, and the correspondence between 3D shapes and their 2D developments, along with geometric transformations such as rotation and translation. ③ Construction highlights: uses commands like Point, Circle, Cylinder, Intersect, Line, OrthogonalLine, Rotate, If, and Segment; objects include t, A, B, c, θ, r, ϕ, b, e, f, C, a, D, g, k, e', d, h, E, F, G, H, m, n, p, q, s, i, l, j; sliders control the unfolding angle and dimensions. ④ Suitable for middle school (upper grades) and high school solid geometry lessons, as well as introductory college geometry or math lab activities, helping students understand the derivation of the cylinder surface area formula and enhance spatial visualization skills.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Point(yAxis) → B
  3. Circle(A, B) → c
  4. Cylinder(c, 3) → b, e, f
  5. Intersect(zAxis, e) → C,
  6. Line(C, xAxis) → a
  7. OrthogonalLine(D, xOyPlane) → g
  8. Circle(g, C) → k
  9. Rotate(Rotate[e, ϕ, a], ϕ, xAxis) → e'
  10. Rotate(Rotate[e, ϕ, a], ϕ, xAxis) → d
  11. Circle(A, B) → h
  12. If(t < 1, ((r * sin((-θ))), ((r * (1 - cos((-θ)))) * cos(ϕ)) - (3 * sin(ϕ)), (3 * cos(ϕ)) + ((r * (1 - cos((-θ)))) * sin(ϕ))), ((-π), (-3 * sin(ϕ)), (3 * cos(ϕ)))) → E
  13. If(t < 1, ((r * sin(θ)), ((r * (1 - cos(θ))) * cos(ϕ)) - (3 * sin(ϕ)), (3 * cos(ϕ)) + ((r * (1 - cos(θ))) * sin(ϕ))), (π, (-3 * sin(ϕ)), (3 * cos(ϕ)))) → F
  14. If(t < 1, ((r * sin((-θ))), ((r * (1 - cos((-θ)))) * cos(ϕ)), ((r * (1 - cos((-θ)))) * sin(ϕ))), (-3.141592653589793, 0, 0)) → G
  15. If(t < 1, ((r * sin(θ)), ((r * (1 - cos(θ))) * cos(ϕ)), ((r * (1 - cos(θ))) * sin(ϕ))), (3.141592653589793, 0, 0)) → H
  16. Segment(E, G) → m
  17. Segment(F, H) → n
  18. Line((0, 1, 0), zAxis) → p
  19. Circle(p, B) → q
  20. Circle(p, B) → s
  21. If(t < 1, Surface[(r * sin((u * θ))), ((r * (1 - cos((u * θ)))) * cos(ϕ)) - (v * sin(ϕ)), (v * cos(ϕ)) + ((r * (1 - cos((u * θ)))) * sin(ϕ)), u, -1, 1, v, 0, 3], Surface[(π * u), ((-v) * sin(ϕ)), (v * cos(ϕ)), u, -1, 1, v, 0, 3]) → i
  22. If(t < 1, CurveCartesian[(r * sin((u * θ))), ((r * (1 - cos((u * θ)))) * cos(ϕ)), ((r * (1 - cos((u * θ)))) * sin(ϕ)), u, -1, 1], CurveCartesian[(π * u), 0, 0, u, -1, 1]) → l
  23. If(t < 1, CurveCartesian[(r * sin((u * θ))), ((r * (1 - cos((u * θ)))) * cos(ϕ)) - (3 * sin(ϕ)), (3 * cos(ϕ)) + ((r * (1 - cos((u * θ)))) * sin(ϕ)), u, -1, 1], CurveCartesian[(π * u), (-3 * sin(ϕ)), (3 * cos(ϕ)), u, -1, 1]) → j

AI Drawing Prompt Reference

Draw a cylinder and dynamically unfold its surface into a flat net. The cylinder has two circular bases and a curved lateral surface, which unfolds into a rectangle while the bases become two circles. Use sliders to control the unfolding angle and dimensions, showing the transformation from 3D to 2D. Include the cylinder's axis, base circles, the rectangle boundary of the unfolded lateral surface, and the curves and surfaces formed during unfolding. This visualizes the geometric transformation and aids in understanding the derivation of the cylinder's surface area formula.
The prompt above was compiled from the original drawing commands and is for reference only.