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Curved Surface Area of Cones by Anthony OR 柯志明 (2)

Curved Surface Area of Cones by Anthony OR 柯志明 (2)
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2014-12-05 15:25
CircleTrigonometrySolid GeometryGeometric Construction
In collections Area, Surface Area, Volume, Cross Section, 3D 3D Geometry & Cross Sections Volumes and Surface Areas of Pyramids, Cones and Spheres GCSE Higher Ch7 Area and Volume Geometry: Surface Area & Volume Geometry: Surface Area & Volume Geometry: Surface Area & Volume GA Chapter 11: Surface Area and Volume Volumes and Surface Areas of Pyramids, Cones and Spheres More about Volumes and Surface Areas Connolly Geometry Volume Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes and Surface Areas of Pyramids, Cones and Spheres
This example demonstrates how the curved surface of a cone can be flattened into a sector, dissected, and rearranged to show its area is πrl. It involves core concepts like surface area, cone geometry, Pythagorean theorem, and trigonometry. Constructed using commands like Point, If, Sequence, CircleArc, Segment, Midpoint, Circle, Intersect, Rotate, Surface, CircumcircleArc, Vector, and Angle. Ideal for high school mathematics education and geometric learning.

GeoGebra Commands & Parameters

  1. Point(CircleArc[(0, 1), (1, 1), (0, 0)]) → Q
  2. If(y(Q) > 0, CircleArc[(0, 1), (0, 0), Q], 6.283185307179586) → s
  3. If(y(Q) > 0, Angle[(0, 0), (0, 1), Q], 6.283185307179586) → α'
  4. Sequence((sin((i * θ)), 1 - cos((i * θ))), i, 0, (2 * n)) → list1
  5. If(t ≤ 1, t, 1) → k
  6. If(t ≥ 1, t - 1, 0) → k'
  7. Sequence(If[k < 1, (x(O_1) + ((l * sin((i * φ))) * cos((i * φ)) / sin(φ)), y(O_1) + ((l * sin((i * φ))) * sin((i * φ)) / sin(φ))), (x(O_1) + (l * i), y(O_1))], i, 0, (2 * n)) → list2
  8. Sequence(Sector[Rotate[Translate[Circle[(x(O_1) + 0, y(O_1) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]]], i, 1, n) → list3
  9. CircleArc((0, 1 / (1 - m)), O_1, L) → b
  10. Segment((0, 0), (s, 0)) → d
  11. Midpoint(d) → B'
  12. CircleArc((0, 1), Q, (0, 0)) → c
  13. Segment(O, (x(O), y(O) - 1)) → e
  14. Midpoint(e) → G
  15. Circle((0, 1), 1) → g
  16. Midpoint(Q, O) → H
  17. Segment(O, A) → a
  18. Circle(P, 1, yAxis) → p
  19. Intersect(xOyPlane, p) → D, C
  20. Rotate(O_1, α, a) → F
  21. Surface((u * cos(v)), (u * sin(v)) + 1, f(u), u, 0, abs(x(D)), v, -1.5708, (3 * π / 2) - (2 * γ)) → o
  22. Rotate(D_1, α, a) → D'
  23. CircumcircleArc(D_1, C, D') → q
  24. Circle(O, D_1, a) → c_1
  25. Segment(P, D') → i
  26. Segment(P, D_1) → a_1
  27. Point(b) → J
  28. Segment(O, D_1) → b_1
  29. Vector(A + (0, 0, 0.3), A + (0, 0, 0.06)) → u
  30. Angle(D_1, O, A) → β
  31. Sequence(Sector[Rotate[Rotate[Translate[Circle[(x(O_1) + 0, y(O_1) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O_1) + (n * l) - sin(θ / 2) / 2, y(O_1) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O_1) + (n * l) - sin(θ / 2) / 2, y(O_1) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O_1) + (n * l) - sin(θ / 2) / 2, y(O_1) + cos(θ / 2) / 2)]], i, n + 1, (2 * n)) → list4

AI Drawing Prompt Reference

Draw a cone's lateral surface area by first creating a circle centered at (0,1), then selecting a point Q on the circle and drawing an arc s. Next, construct an angle α'. Then, create a sequence of n points list1 and another sequence list2 based on k and k'. Using list1 and list2, rotate and translate to form a series of sectors, ultimately creating the unfolded lateral surface of the cone.
The prompt above was compiled from the original drawing commands and is for reference only.