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Curved Surface Area of Cones by Anthony OR 柯志明

Curved Surface Area of Cones by Anthony OR 柯志明
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2014-09-28 12:21
CircleSolid GeometryGeometric Construction
In collections Surface Area and Volume Surface Area and Volume Surface Area and Volume
This example demonstrates how to calculate the curved surface area of a cone with base radius r and slanted height l. It involves core math concepts like cone geometry, π, and surface area formulas. GeoGebra commands/objects used include Point, If, Sequence, Segment, Min, Max, CircleArc, Midpoint, CircleSector, Circle for constructing the cone model and computing the surface area. Suitable for high school mathematics teaching and geometry learning, helping students understand the principles of surface area calculation.

GeoGebra Commands & Parameters

  1. Point(CircleArc[(0, 1), (1, 1), (0, 0)]) → A
  2. If(y(A) > 0, Angle[(0, 0), (0, 1), A], 6.283185307179586) → α
  3. Sequence((sin(i * θ), 1 - cos(i * θ)), i, 0, 2 * n) → list1
  4. Segment(E, (x(E) + 1, y(E))) → a
  5. Point(a) → F
  6. Min(2 * (x(F) - x(E)), 1) → k
  7. Max(2 * (x(F) - x(E)) - 1, 0) → k'
  8. Sequence(If[k < 1, (x(O) + l * sin(i * φ) * cos(i * φ) / sin(φ), y(O) + l * sin(i * φ) * sin(i * φ) / sin(φ)), (x(O) + l * i, y(O))], i, 0, 2 * n) → list2
  9. Sequence(Sector[Rotate[Translate[Circle[(x(O) + 0, y(O) + 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
  10. Sequence(Sector[Rotate[Rotate[Translate[Circle[(x(O) + 0, y(O) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], -(2 * i - 1) * k * θ / 2, Element[list2, i]], 180° * k', (x(O) + n * l - sin(θ / 2) / 2, y(O) + 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) + n * l - sin(θ / 2) / 2, y(O) + 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) + n * l - sin(θ / 2) / 2, y(O) + cos(θ / 2) / 2)]], i, n + 1, 2 * n) → list4
  11. If(y(A) > 0, CircleArc[(0, 1), (0, 0), A], 6.283185307179586) → s
  12. CircleArc((0, 1 / (1 - m)), O, L) → b
  13. Segment((0, 0), (s, 0)) → d
  14. Midpoint(d) → B'
  15. CircleArc((0, 1), A, (0, 0)) → c
  16. Midpoint(a) → C
  17. Point(Segment[(0, 1), (s, 1)]) → D
  18. Segment(D, (x(D), y(D) - 1)) → e
  19. Midpoint(e) → G
  20. CircleSector((0, 1), (0, 0), A) → f
  21. Circle((0, 1), 1) → g
  22. Midpoint(A, D) → H