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Visualizing Curved Surface Area of Cones by Dan Dale

Visualizing Curved Surface Area of Cones by Dan Dale
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2019-07-01 21:35
TrigonometrySolid GeometryGeometric Construction
This example demonstrates how the curved surface of a cone can be flattened into a sector, which is then dissected and rearranged to show that its area is πrl. Key math concepts include surface area, sector area, cone geometry, and trigonometric applications. It uses commands like Point, If, Sequence, CircleArc, Segment, Midpoint, Circle, Intersect, Rotate, Surface, CircumcircleArc, Vector, and Angle in GeoGebra. Suitable for high school students learning solid geometry and geometric constructions, as well as for mathematical demonstrations and teaching purposes.

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