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Curved Surface Area of Cones by Nicola Trubridge

Curved Surface Area of Cones by Nicola Trubridge
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2018-03-25 23:56
VectorSolid GeometryGeometric Construction
This example demonstrates how the curved surface area of a cone is equal to the area of a sector. By using rotation and mirroring, the cone's curved surface can be unfolded into a sector. The construction involves commands such as Point, Circle, Intersect, Rotate, Mirror, Segment, CircumcircleArc, Surface, and Vector. Suitable for middle school and above students to learn solid geometry.

GeoGebra Commands & Parameters

  1. Point(zAxis) → A
  2. Point(Segment[A, (0, 0, 0)]) → P
  3. Point(xAxis) → B
  4. Circle(P, Distance[A, B], yAxis) → c
  5. Intersect(xOyPlane, c) → C, D
  6. Intersect(xAxis, yAxis) → O
  7. Rotate(D, γ, zAxis) → D_1
  8. Rotate(D_1, α, zAxis) → D'
  9. Rotate(D_1, (π * x(B) / x(D)), zAxis) → D'_1
  10. Mirror(D'_1, O) → E
  11. Segment(P, D') → e
  12. Circle(zAxis, B) → g
  13. CircumcircleArc(D_1, E, D') → h
  14. CircumcircleArc(D_1, D'_1, D') → d
  15. Segment(P, D_1) → b
  16. Surface((u * cos(v)), (u * sin(v)), f(u), u, 0, x(D), v, γ, (2 * π) - γ) → a
  17. Vector(A + (0, 0, 1), A + (0, 0, 0.2)) → u
  18. Segment(O, A) → i
  19. Segment(O, B) → j

AI Drawing Prompt Reference

Draw the lateral surface of a cone unfolded into a sector. Start by creating point A on the z-axis and point B on the x-axis. Create a circle c with A and B as centers and AB as the radius on the xOy plane, and find the intersection points C and D of circle c with the x-axis. Then rotate point D around the z-axis to form D_1, and rotate D_1 around the z-axis to form D'. Next, mirror D'_1 at the origin O to form E. Connect point P and D' to form line segment e, create a circle g with B as the center, and connect D_1, E, and D' to form arc h, and connect D_1, D'_1, and D' to form arc d. Finally, use parameters u and v to create a surface a, representing the unfolded lateral surface of the cone.
The prompt above was compiled from the original drawing commands and is for reference only.