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Unfolding the Curved Surface of a Cone

Unfolding the Curved Surface of a Cone
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2018-03-09 06:11
Solid GeometryCalculusGeometric Construction
This example demonstrates how the curved surface of a cone can be flattened into a sector, and how rearranging the sector helps understand its area formula πrl. It involves concepts of cones, sectors, and triangles, and uses GeoGebra commands such as points, arcs, and vectors for construction. Suitable for middle school and above students learning solid geometry and area calculation.

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. 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
  10. CircleArc((0, 1 / (1 - m)), O_1, L) → b
  11. Segment((0, 0), (s, 0)) → d
  12. Midpoint(d) → B'
  13. CircleArc((0, 1), Q, (0, 0)) → c
  14. Segment(O, (x(O), y(O) - 1)) → e
  15. Midpoint(e) → G
  16. Circle((0, 1), 1) → g
  17. Midpoint(Q, O) → H
  18. Segment(O, A) → a
  19. Circle(P, 1, yAxis) → p
  20. Intersect(xOyPlane, p) → D, C
  21. Rotate(O_1, α, a) → F
  22. Surface((u * cos(v)), (u * sin(v)) + 1, f(u), u, 0, abs(x(D)), v, -1.5708, (3 * π / 2) - (2 * γ)) → o
  23. Rotate(D_1, α, a) → D'
  24. CircumcircleArc(D_1, C, D') → q
  25. Circle(O, D_1, a) → c_1
  26. Segment(P, D') → i
  27. Segment(P, D_1) → a_1
  28. Point(b) → J
  29. Segment(O, D_1) → b_1
  30. Vector(A + (0, 0, 0.3), A + (0, 0, 0.06)) → u
  31. Angle(D_1, O, A) → β