圆锥表面积公式：用展开图观察与推导

GeoGebra 命令及参数

1. Shuffle(Sequence[5]) → possibleRads, {1, 4, 3, 2, 5}
2. If(countQuestion ≟ 0, 2, Element[possibleRads, Mod[countQuestion, Length[possibleRads]] + 1]) → r1
3. Shuffle(KeepIf[1 ≤ sqrt(kk^(2) - r1^(2)) ≤ 5, kk, Sequence[7]]) → possibleHsl, {3, 5, 4}
4. If(countQuestion ≟ 0, 5, possibleHsl(1)) → Hsl
5. Point(yAxis) → CA
6. Point(yAxis) → CB
7. Ray(A, (x(A), y(A), 40)) → h
8. Point(yAxis) → B
9. Point(Segment[(0, 0, 0.1), (0, 0, 4)]) → CD
10. Random(3, 8) → nRand
11. Circle(CA, CB) → d
12. Segment((1, 0, 0), (1, 0, 5)) → aCon
13. Circle(aCon, (0, 0, 0)) → cCon
14. Intersect(xAxis, aCon) → KB
15. Cone(KB, KA, 1) → bCon, dCon, eCon
16. Intersect(yAxis, xAxis) → KC
17. Segment(KC, KA) → lCon
18. Distance(KB, KC) → rCon
19. Angle(KA, KC, KB) → α
20. Rotate(KA, (abs(tCil) * α), yAxis) → KAA
21. OrthogonalLine(KAA, xOyPlane) → fCon
22. Intersect(fCon, xOyPlane) → KBB
23. Rotate(KC, bet / 2, fCon) → KCC
24. Rotate(KC, ((-bet)) / 2, fCon) → KCR
25. CircumcircleArc(KCC, KC, KCR) → gCon
26. Distance(KAA, KBB) → hCon
27. Distance(KBB, KCC) → rcCon
28. Surface(((kCon - ((v * rcCon) * cos((u * bet / 2)))) * cos(ϕCon)) - (((1 - v) * hCon) * sin(ϕCon)), ((v * rcCon) * sin((u * bet / 2))), (((1 - v) * hCon) * cos(ϕCon)) + ((kCon - ((v * rcCon) * cos((u * bet / 2)))) * sin(ϕCon)), u, -1, 1, v, 0, 1) → jCon
29. CurveCartesian(((kCon - (rcCon * cos((u * bet / 2)))) * cos(ϕCon)), (rcCon * sin((u * bet / 2))), ((kCon - (rcCon * cos((u * bet / 2)))) * sin(ϕCon)), u, -1, 1) → mCon
30. Segment(KE, KF) → nCon
31. Segment(KD, KF) → pCon
32. Corner(5) → SizeGV1
33. Vector((1; (pi / 2 * tCil2) + (pi / 6 * (1 - tCil2)); ((-pi)) / 2 + ((0.8 * pi / 2) * (1 - tCil2)))) → view
34. Rotate(jCon, -90°) → jCon'
35. Rotate(KE, -90°) → KE'
36. Rotate(KF, -90°) → KF'
37. Segment(KE', KF') → nCon'
38. Rotate(KD, -90°) → KD'
39. Rotate(KF, -90°) → KF'_{1}
40. Segment(KD', KF'_{1}) → pCon'
41. Rotate(cCon, -90°) → cCon'
42. Rotate(mCon, -90°) → mCon'
43. Rotate(dCon, -90°) → dCon'
44. If(r1 / Hsl ≤ 1 / 2, 1, 2) → factor
45. Translate(KE', Vector[(0, -1 + (2 * tCil2), 0)]) → EP1
46. Translate(KF', Vector[(0, -1 + (2 * tCil2), 0)]) → EP2
47. Vector(Midpoint[EP1, EP2], EP1) → u
48. Vector(Midpoint[EP1, EP2], EP2) → v
49. Vector(Midpoint[EPr1, EPr2], EPr2) → w
50. Vector(Midpoint[EPr1, EPr2], EPr1) → a
51. Segment((0, -1, 0), (0, 0)) → f
52. Segment(EP1, KE') → g
53. Segment(EP2, KF') → i
54. Segment(EPr2, (0, -1, 0)) → j
55. Segment(EPr1, (0, 0, 0)) → k
56. Text(r1, (0, 0), true, true) → labelRToMeasure
57. Text(Hsl, (0, 0), true, true) → labelLToMeasure
58. Vector(Corner[labelRToMeasure, 1], Corner[labelRToMeasure, 3]) → measureLabelR
59. Vector(Corner[labelLToMeasure, 1], Corner[labelLToMeasure, 3]) → measureLabelL
60. Text(r1, Midpoint[EPr1, EPr2] + ((-x(measureLabelR)), y(measureLabelR) / 2), true, true) → labelR
61. Text(Hsl, Midpoint[EP1, EP2] + (((-tCil2) * x(measureLabelL) / 4), (((2 * tCil2) - 1) * y(measureLabelL)), (y(measureLabelL) * (1 - tCil2))), true, true) → labelL