圆的面积：通过几何变换与积分计算

GeoGebra 命令及参数

1. Sequence(CircleSector[(0, 0), Rotate[(2, 0), ((2 * i) * pi / n), (0, 0)], Rotate[(2, 0), ((2 * (i + 1)) * pi / n), (0, 0)]], i, 0, n, 1) → l1
2. Sequence(Element[l1, i], i, 1, n / 4, 1) → topr
3. Sequence(Element[l1, i], i, n / 4 + 1, (2 * n / 4), 1) → topl
4. Sequence(Element[l1, i], i, (2 * n / 4) + 1, (3 * n / 4), 1) → bottoml
5. Sequence(Element[l1, i], i, (3 * n / 4) + 1, n, 1) → bottomr
6. Sequence((0, 0), i, 1, n / 4, 1) → l2
7. Sequence((0, ((4 * i / n) * pi)), i, 1, n / 4, 1) → l3
8. Sequence(Dilate[(0, 0), If[0 < a ≤ 1, 1 - a, 0], Element[l3, i]], i, 1, n / 4, 1) → l4
9. Mirror(l4, (0, 0)) → l4'
10. Sequence(Vector[(0, 0), Element[l4, i]], i, 1, n / 4, 1) → l5
11. Sequence(Translate[Element[topr, i], Vector[Element[l5, i]]], i, 1, n / 4, 1) → l6
12. Sequence(Rotate[Element[l6, n / 4 - i], (-(pi / 2 - (((2 * i) + 1) * pi / n))), Element[l4, n / 4 - i]], i, 0, n / 4 - 1, 1) → l7
13. Sequence(Rotate[Element[l6, n / 4 - i], (-(pi / 2 - (((2 * i) + 1) * pi / n))), Element[l4, n / 4 - i]], i, 0, n / 4 - 1, 1) → l8
14. Sequence(Rotate[Element[l6, n / 4 - i], If[0 < a ≤ 1, ((-(pi / 2 - (((2 * i) + 1) * pi / n))) * a), (-(pi / 2 - (((2 * i) + 1) * pi / n)))], Element[l4, n / 4 - i]], i, 0, n / 4 - 1, 1) → l9
15. Mirror(l9, yAxis) → l9'
16. Mirror(l9, xAxis) → l9'_{1}
17. Mirror(l9', xAxis) → l9''
18. Distance((0, 0), Element[l4, 1]) → b
19. Dilate((0, 0), If[1 < a ≤ 2, 2 - a, 0], Element[l4, 1]) → H
20. Vector((0, 0), H) → u
21. Translate(l9'_{1}, u) → l10
22. Translate(l9'', u) → l9'''
23. Point(yAxis) → A
24. Circle(A, 2) → c
25. Ray(A, A - (1, 0)) → f
26. Intersect(c, f, 1) → I
27. Rotate(I, ((-pi)) / n, A) → I'
28. Dilate(A, If[2 < a ≤ 3, 3 - a, 0], I') → A'
29. Vector(A, A') → v
30. Translate(l9, v) → l9'_{2}
31. Translate(l10, v) → l10'
32. Translate(topr, Vector[(4, 0)]) → topr'
33. Translate(topl, Vector[(4, 0)]) → topl'
34. Translate(bottoml, Vector[(4, 0)]) → bottoml'
35. Translate(bottomr, Vector[(4, 0)]) → bottomr'
36. Translate(eq1, Vector[(4, 0)]) → eq1'
37. RotateText(text1, pi / 2) → text3