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Cusanus π Approximation: Polygon Inscribed in Circle

Cusanus π Approximation: Polygon Inscribed in Circle
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2023-03-14 09:52
TriangleCirclePolygonGeometric ConstructionLocus
In collections Pi Approximations
This GeoGebra example demonstrates the geometric method of approximating π proposed by 15th century philosopher Nicholas of Cusa (Cusanus). By inscribing equilateral triangles in a circle and progressively increasing the number of polygon sides, the perimeter of the inscribed polygon approaches the circumference of the circle, yielding an approximation of π. Key mathematical concepts include circle circumference formula, limit theory, and polygon approximation. Construction uses: Polygon for inscribed polygons, Circle for auxiliary circles, LineBisector for perpendicular bisectors to find the center, Segment for polygon edges, Midpoint for edge midpoints, Circumference for comparing perimeters. Suitable for high school mathematics extensions, calculus limit demonstrations, and math history electives.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → Vieleck1, a, b, c, d, C_1, D_1
  2. Circle(A, B, C_1) → e
  3. Center(e) → M
  4. LineBisector(a) → f
  5. Intersect(e, f, 1) → C_2
  6. Segment(A, C_2) → g
  7. Segment(C_2, B) → h
  8. Intersect(f, a) → G
  9. Circle(M, G) → k
  10. Midpoint(g) → H
  11. Midpoint(h) → D
  12. Segment(H, D) → i
  13. Segment(M, A) → j
  14. Segment(M, B) → l
  15. Segment(M, C_2) → m
  16. Segment(M, D) → n_{1}
  17. Segment(M, H) → p
  18. Intersect(i, m) → C
  19. Circle(M, C) → q
  20. Circle(M, D) → r
  21. Intersect(k, b, 1) → E
  22. Segment(M, E) → s
  23. Segment(M, C) → t
  24. Circumference(q) → umfangq
  25. PointIn(q) → Punktq_{1}
  26. Circumference(r) → umfangr
  27. PointIn(r) → Punktr
  28. Circumference(k) → umfangk
  29. PointIn(k) → Punktk_{1}
  30. Circumference(q) → o
  31. PointIn(q) → Punktq
  32. Circumference(k) → u
  33. PointIn(k) → Punktk

AI Drawing Prompt Reference

Draw a square inscribed in a circle, then gradually increase the number of sides of the inscribed polygon to see how its perimeter approximates the circumference of the circle. First, draw a circle with center M and radius MA, then draw line segment ME through M and the intersection of circle k and circle e. Next, draw a circle with center M and radius ME, then draw line segment MC through M and the intersection of circle r and circle k. Then, draw a circle with center M and radius MC, and finally draw line segment MD through M and the intersection of circle q and circle k, resulting in a square inscribed in the circle.
The prompt above was compiled from the original drawing commands and is for reference only.