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Net of a Cylinder by danielhenry

Net of a Cylinder by danielhenry
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2020-02-22 06:04
CircleSolid GeometryTransformationGeometric ConstructionSlider Animation
① This GeoGebra example dynamically demonstrates the 3D cylinder model and its unfolding into a 2D net, visually showing that the surface area consists of two circular bases and one rectangular lateral face. ② Core math concepts include solid geometry, cylinder nets, surface area calculation, and the relationship between 3D and 2D representations via rotation and projection. ③ Construction highlights: uses commands such as Point, Circle, Cylinder, Intersect, OrthogonalLine, Rotate, If, and Segment; key 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; slider θ controls the unfolding angle, while r and ϕ set base radius and position. ④ Suitable for middle/high school geometry lessons, introductory solid geometry, and interactive math visualization, helping students grasp the derivation of the cylinder's surface area formula.

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's net, consisting of two circular bases and one rectangular side face. Start by creating two points A and B on the y-axis, then draw a circle c with A and B as centers. Next, create a cylinder b, e, f with c as the base. Find the intersection point C of circle c and the z-axis, then draw a line a parallel to the x-axis through C. Draw a line g perpendicular to the xOy plane and create a circle k with g and C as centers. Rotate e and f around the a axis to form the side net. Use parameters t, r, and ϕ to control the unfolding angle and circle size, and finally draw the various parts of the net using conditional statements.
The prompt above was compiled from the original drawing commands and is for reference only.