Circle Area Formula Demonstration

GeoGebra Commands & Parameters

1. Segment(E, (x(E) + 2, y(E))) → a
2. Point(a) → F
3. Min(x(F) - x(E), 1) → k
4. Point(yAxis) → O
5. Sequence(If[k < 1, (x(O) + ((l * sin((i * φ))) * cos((i * φ)) / sin(φ)), y(O) + ((l * sin((i * φ))) * sin((i * φ)) / sin(φ))), (x(O) + (l * i), y(O))], i, 0, (2 * n)) → list2
6. Sequence((x(O) + sin((i * θ)), y(O) + 1 - cos((i * θ))), i, 0, (2 * n)) → list1
7. Max(x(F) - x(E) - 1, 0) → k'
8. Rotate((x(O) + 0, y(O) + 1), ((-k) * θ / 2), O) → A
9. Rotate(O, ((-k) * θ / 2), O) → O'
10. Rotate(Element[list1, 1], ((-k) * θ / 2), O) → B
11. Sector(Rotate[Circle[(x(O) + 0, y(O) + 1), O], ((-k) * θ / 2), O], Rotate[O, ((-k) * θ / 2), O], Rotate[Element[list1, 1], ((-k) * θ / 2), O]) → c
12. Rotate((x(O) + 0, y(O) + 1), ((-k) * θ / 2), O) → C
13. Rotate(O, ((-k) * θ / 2), O) → O'_1
14. Rotate(Element[list1, 1], ((-k) * θ / 2), O) → D
15. Arc(Rotate[Circle[(x(O) + 0, y(O) + 1), O], ((-k) * θ / 2), O], O'_1, D) → d
16. CurveCartesian(t - sin(t), 1 - cos(t), t, 0, 6.28319) → b
17. Circle((0, 1), 1) → h
18. Ray((0, 0), (1, 0)) → i
19. Point(Segment[(0, 1), (6.28319, 1)]) → H
20. Segment(H, (x(H), 0)) → j
21. Midpoint(j) → I
22. Midpoint(a) → J
23. Circle(L, 1 / (1 - m)) → p
24. Circle((0, 0), 1 / (1 - m)) → q
25. Intersect(p, q, 1) → M
26. Intersect(p, q, 2) → N
27. Circle((0, 1), 1) → c_1
28. Segment((0, 0), (6.283185307179586, 0)) → e
29. Segment(G, (x(G), y(G) - 1)) → r
30. CircleArc((0, 1 / (1 - m)), O, L) → s
31. Sequence(Midpoint[Element[list2, i], Element[list2, i + 1]] + (l / ((2 * tan(θ / 2))) * UnitOrthogonalVector[Vector[Element[list2, i], Element[list2, i + 1]]]), i, 1, (2 * n) + 1) → list5
32. Sequence(CircleSector[Element[list5, i], Element[list2, i], Element[list2, i + 1]], i, 1, n) → list3
33. Sequence(Rotate[CircleSector[Element[list5, i], Element[list2, i], Element[list2, i + 1]], (180° * k'), (((n + 0.75) * l) - (sin(θ / 2) * 2), cos(θ / 2) / 2)], i, n + 1, (2 * n) + 1) → list4