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Bounding π Using Archimedes' Polygon Method

Bounding π Using Archimedes' Polygon Method
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2018-10-24 21:36
CirclePolygonAngleTrigonometryGeometric Construction
This GeoGebra example demonstrates how Archimedes approximated π by inscribing and circumscribing regular polygons around a unit circle, using polygon perimeters to establish upper and lower bounds for π. Key mathematical concepts include limits, trigonometric functions, polygon approximation, and the geometric definition of π. Construction highlights: Iteration command for iterative calculations, Polygon for regular polygons, Circle for the unit circle, Angle for central angle measurements, Textfield for dynamic数值 display, and Slider for adjusting polygon sides n to achieve interactive approximation. Ideal for high school mathematics classrooms teaching limits and approximation concepts, or as a visual aid for university-level analysis courses.

GeoGebra Commands & Parameters

  1. Iteration((sqrt((1 + x(M)) / 2), sqrt((1 - x(M)) / 2)), M, {(sqrt(3) / 2, 1 / 2)}, k) → A
  2. Polygon((1; 0), (1; (2 * π / n)), n) → poly1
  3. Polygon((1, (-tan(π / n))), (1, tan(π / n)), n) → poly2
  4. Textfield(k) → InputBox1
  5. Intersect(xAxis, yAxis) → O
  6. Polygon((1, (-tan(π / 6))), (1, tan(π / 6)), 6) → poly5
  7. Circle(O, C) → c
  8. Polygon((1; 0), (1; (2 * π / 6)), 6) → poly4
  9. Point(c) → D
  10. Segment(D, O) → f
  11. Angle(C, O, D) → α
  12. Segment(O, F) → h
  13. Segment(O, G) → i
  14. Midpoint(D, C) → E
  15. Segment(O, E) → g
  16. Polygon(O, C, D) → t1, d, b_1, c_1
  17. Angle(C, O, E) → γ
  18. Angle(O, E, C) → β
  19. Segment(D, E) → j
  20. Segment(E, C) → l
  21. Polygon(G, F, O) → t2, o, g_1, f_1
  22. Segment(F, C) → m
  23. Segment(C, F) → p
  24. Segment(C, G) → q

AI Drawing Prompt Reference

Draw a unit circle centered at the origin. Construct a regular hexagon inscribed in the circle with one vertex on the positive x-axis, and another regular hexagon circumscribed about the circle with horizontal top and bottom sides. Use a slider to control the number of sides for both inscribed and circumscribed regular polygons, which approach the circle as the side count increases. Show radius segments from the center to polygon vertices, and perpendicular segments from the center to tangent points on the circumscribed polygon. Label central angles, half-central angles, and interior angles of triangles formed by radii and sides. Dynamically display the perimeters of the inscribed and circumscribed polygons to estimate lower and upper bounds for pi.
The prompt above was compiled from the original drawing commands and is for reference only.