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Surface Area of a Tipi

Surface Area of a Tipi
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2017-10-16 21:24
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates how to calculate the surface area of a tipi using geometric construction. Key mathematical concepts include the properties of a cone and the area calculation formula. GeoGebra commands such as point, sector division, arc, and circle are used in the construction. Suitable for middle school and above students or teachers to understand the surface area calculation in solid geometry.

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
  23. Point(e) → J
  24. Angle(J, D, A) → β