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Inscribed and Circumscribed Polygons for Pi Approximation

Inscribed and Circumscribed Polygons for Pi Approximation
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2015-12-11 05:55
CirclePolygonTrigonometryCoordinate GeometryCalculus
Approximate the value of π using the perimeters of inscribed and circumscribed polygons. This example demonstrates the concept of limits and how polygons can approximate the circumference of a circle. It involves core mathematical concepts such as trigonometry, coordinate geometry, and calculus. Constructed using GeoGebra commands like Circle, Rotate, Angle, Polygon, Line, OrthogonalLine, Intersect, and Distance. Suitable for high school and above students to understand the approximation of π.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Rotate(B, (360 * ° / n), A) → B'
  3. Angle(B, A, B') → α
  4. Polygon(B, B', n) → poly1, b, I, b', a, e, k, q, C_1, D, F
  5. Line(A, B) → g
  6. OrthogonalLine(B, g) → h
  7. Line(A, B') → i
  8. OrthogonalLine(B', i) → j
  9. Intersect(h, j) → C
  10. Line(A, C_1) → l
  11. OrthogonalLine(C_1, l) → m
  12. Intersect(j, m) → C'
  13. Polygon(C, C', n) → poly2, c_1, J, c', d, f, p, r, G, E, H
  14. Distance(B, B') → distanceBB'
  15. Distance(C, C') → distanceCC'
  16. Distance(A, B) → distanceAB

AI Drawing Prompt Reference

Draw a circle c, then rotate point B around point A to form a polygon poly1 with n sides. Construct the perpendicular line h from point C1 to the tangent line i, and find their intersection point C. Draw a line l from point A through C1 and its perpendicular line m, and find their intersection point C'. Form another polygon poly2 with n sides using points C and C'. Measure the distances between B and B', C and C', and A and B, and observe how the perimeter of the polygon approximates the circumference of the circle to estimate the value of π.
The prompt above was compiled from the original drawing commands and is for reference only.