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UNIT 11 LESSON 2: VOLUME OF A CONE

UNIT 11 LESSON 2: VOLUME OF A CONE
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2020-12-15 00:47
CirclePolygonSolid GeometryGeometric ConstructionSlider Animation
① This interactive GeoGebra applet is designed for pupils to review the volume formula of a cone, perform related calculations, and observe the net of a cone. ② Core mathematical concepts include the cone volume formula V = (1/3)πr²h, and the geometric relationship between the cone's lateral surface (a sector) and its circular base, involving parameters such as radius, height, slant height, and central angle. ③ Construction highlights: based on the construction summary, the applet uses commands such as If, Conic, Segment, Midpoint, Point, Distance, Rotate, CircleSector, Polygon, Intersect, Arc, Circumference, Angle, CircumcircleArc, and Textfield to create the cone's base circle, lateral sector, slant height, height, volume calculation text field, and adjustable sliders (e.g., h, r, θ_d), while dynamically generating the net and related objects. ④ Suitable for middle or high school students learning solid geometry, particularly for intuitively understanding the derivation of the cone volume formula and the correspondence between the net and the 3D shape; also useful for classroom demonstrations or self-paced exploration.

GeoGebra Commands & Parameters

  1. If(net < -1, (((-net) - 1) * 3), 0) → net_θ
  2. If(net < -1, (3 * ((-net) - 1)) + 0.2, 0.2) → net_b
  3. If(net < -1, -1, net) → net_e
  4. If(net_e > -0.001, 0, (-net_e)) → net_h
  5. Conic(A, B, C, D, E) → c
  6. If(net > 0, 1, 0) → net_1
  7. Conic(A, B, C, D, E) → d
  8. Segment(V, A') → b
  9. Segment(V, F') → e
  10. Midpoint(A', F') → M_{A'F'}
  11. Point(c) → G
  12. Conic(A', B', C', F', E') → k
  13. Conic(A', B', C', F', E') → p
  14. Distance(O, V) → distanceOV
  15. Segment(O_c, O_u) → l
  16. Rotate(O_o, (θ * 2), V) → H
  17. Rotate(O_o, ((-θ) * 2), V) → H'
  18. CircleSector(V, H', H) → q
  19. Polygon(V, A', F') → poly1, f', v, a
  20. Distance(V, H') → distanceVH'
  21. Segment(V, H) → b'
  22. Segment(V, H') → e'
  23. Intersect(p, b') → K, K_2
  24. Intersect(p, e') → J, J_2
  25. Segment(V, J_2) → g
  26. Segment(V, K_2) → i
  27. Arc(p, A', K_2) → t
  28. Arc(p, J_2, F') → f
  29. If(θ + (0.6 * θ_d) < (((2 * π) * r / l_s) * 0.5), θ + (0.6 * θ_d), (((2 * π) * r / l_s) * 0.5)) → θ_c
  30. Rotate(O_o, θ_c, V) → H_1
  31. Rotate(O_o, (-θ_c), V) → H'_1
  32. CircleSector(V, H'_1, H_1) → c_2
  33. Circumference(c_2) → j
  34. Circumference(c) → circumferencec
  35. Distance(V, H'_1) → distanceVH'1
  36. Angle(H'_1, V, H_1) → α
  37. Point(c_2) → I
  38. CircumcircleArc(H', O_o, H) → c_1
  39. CircleSector(V, H'_1, H_1) → s
  40. Segment(M_{A'F'}, F') → m
  41. Segment(M_{A'F'}, V) → n
  42. Angle(F', M_{A'F'}, V) → β
  43. Midpoint(b) → M_{VA'}
  44. Textfield(r) → inputBox1
  45. Textfield(h) → inputBox2

AI Drawing Prompt Reference

Draw a cone with adjustable radius r, height h, and central angle θ using sliders. Show the volume formula V=(1/3)πr²h. Include the base circle, lateral sector (the unfolded cone), and label the radius, height, slant height, and central angle. Dynamically generate the net (unfolded shape) of the cone, labeling relevant angles and edges. Add a text field next to the diagram to display the volume calculation result and explain the relationship between the lateral sector and the base circle after unfolding.
The prompt above was compiled from the original drawing commands and is for reference only.