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Curved Surface Area of Cones by Anthony OR 柯志明 (3)

Curved Surface Area of Cones by Anthony OR 柯志明 (3)
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2014-10-04 17:50
VectorSolid GeometryGeometric Construction
In collections Surface Area and Volume Surface Area and Volume Surface Area and Volume
This example demonstrates how the curved surface area of a cone is equal to the area of a sector. By adjusting the height and base radius of the cone, you can observe changes in the curved surface area. Key mathematical concepts include the geometry of a cone and the calculation of sector area. GeoGebra commands used include Segment, Point, Distance, Sequence, Rotate, CircleArc, Circle, Vector, and Cone. Suitable for middle school and above students learning solid geometry.

GeoGebra Commands & Parameters

  1. Segment(O, H) → a
  2. Point(a) → A
  3. Distance(O, A) → OA
  4. Distance(O, B) → OB
  5. Sequence((r; γ + (β * i / n)), i, 0, n) → list1
  6. Sequence(Polygon[A, Element[list1, i], Element[list1, i + 1]], i, 1, Length[list1] - 1) → list2
  7. Segment(O, A) → b
  8. Rotate(B, γ, a) → B'
  9. Rotate(B, (-γ), a) → B'_1
  10. Segment(A, B'_1) → c
  11. Segment(A, B') → d
  12. CircleArc(O, B', B'_1, xOyPlane) → e
  13. Circle(O, coneRadius, xOyPlane) → f
  14. CircleArc(O, B'_1, B') → g
  15. Segment(O, B') → i
  16. Vector(H + (0, 0, 1), H + (0, 0, 0.2)) → u
  17. Rotate(B', β / 2, a) → B''
  18. Cone(O, H, i) → j, k, l

AI Drawing Prompt Reference

Draw a cone with its lateral surface unrolled into a sector to demonstrate that the curved surface area of the cone equals the area of the sector. Adjust the height and base radius of the cone to observe changes in the sector's area. Key elements include the cone's vertex, base center, and points on the base circle, with the line connecting the vertex and base center representing the cone's height. Initially, sketch the cone's base and lateral surface, then unroll the lateral surface into a sector. The sector's arc length matches the cone's base circumference, and its radius equals the cone's slant height. Finally, dynamically adjust the cone's parameters to observe changes in the sector's area.
The prompt above was compiled from the original drawing commands and is for reference only.