圆的周长与面积公式推导

GeoGebra 命令及参数

1. Sequence((sin(2 * π * k / n), -cos(2 * π * k / n)), k, 0, n, 1) → vertices
2. Polygon(vertices) → regpoly, , , ,
3. Segment((0, 0), Midpoint[Element[vertices, 1], Element[vertices, 2]]) → a
4. Perimeter(regpoly) → p
5. Sequence(Segment[(0, 0), Element[vertices, k]], k, 1, n) → trisides
6. Area(regpoly) → A
7. Circle(O, 1) → c
8. Segment(O, B) → r
9. Sequence(Circle[O, j / k], j, 1, k, 1) → circles_0
10. Sequence(O + (0, (-j) / k), j, 1, k, 1) → initial
11. Zip(Circle[C, P], P, initial) → circles
12. Zip(Rotate[P, 360° * Distance[O, P] / Radius[S], C], P, initial, S, circles) → final
13. Zip(Ray[C, P], P, final) → rays
14. Sequence(Intersect[Element[circles, j], Element[rays, j + 1]], j, 1, k - 1, 1) → final_2
15. Zip(CircleArc[C, P, P'], P, initial, P', final_2) → arcs
16. CircleArc(C, O, Rotate[O, 360° * Distance[O, Element[initial, 1]] / Radius[Element[circles, 1]], C]) → arc_0
17. Sequence(Segment[Element[final, j + 1], Element[final_2, j]], j, 1, k - 1, 1) → endseg
18. CircleArc(C, B, Rotate[B, 360° * Distance[O, B] / Radius[Element[circles, k]], C]) → arc_k
19. Sequence(Element[initial, j] + (2 * π * Radius[Element[circles_0, j + 1]], 0), j, 1, k - 1, 1) → finalh_2
20. Sequence(Segment[Element[initial, j], Element[finalh_2, j]], j, 1, k - 1, 1) → bases
21. Segment(O, O + (2 * π * Radius[Element[circles_0, 1]], 0)) → base_0
22. Segment(B, B + (2 * π * Radius[Element[circles_0, k]], 0)) → base_k
23. Zip(P + (2 * π * Radius[S], 0), P, initial, S, circles_0) → finalh
24. Sequence(Segment[Element[finalh_2, j - 1], Element[finalh, j]], j, 2, k, 1) → heights
25. Segment(finalh_0, Element[finalh, 1]) → height_0
26. Intersect(arc_0, Element[rays, 1]) → final_0,
27. Segment(final_0, Element[final, 1]) → endseg_0
28. Point(Circle[D, 6.283185307179586]) → E
29. Segment(D, E) → e
30. Sequence(1 / k * 2 * π * Radius[Element[circles_0, j]], j, 1, k) → areas
31. Sum(areas) → area