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Polygon Limits: Approaching Circle Perimeter and Area via Inscribed and Circumsc

Polygon Limits: Approaching Circle Perimeter and Area via Inscribed and Circumsc
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2019-09-24 20:10
CirclePolygonSequenceLimitSlider Animation
This GeoGebra example dynamically demonstrates the concept of limits by showing how the perimeter and area of regular polygons with increasing numbers of sides converge to those of a circle. It uses both inscribed and circumscribed polygons to bracket the circle's circumference and area, illustrating the classic method for approximating pi. Core mathematical concepts include limits, regular polygons, inscribed and circumscribed circles, apothem, sector and segment areas, and ratio comparisons. The construction employs GeoGebra commands such as Sequence, Element, Circle, Circumference, Segment, Sum, Area, If, Textfield, TableText, and First. Key objects include pointList1, segList, perimeterPoly, midPtList, apothemList, radius_2, circle_2, circum_2, radius_1, radiiList, circle_1, showRadii, showApothem, showMidPoints, sectorList, apothArea, segmentArea, showSector, valueSegAreaTotal, areaPolygon, areaPolyText, arclist, areaCircle_1, sumSegCirText, showSegment, areaCircleText, ratioPolyArea2Circle_1, ratioCircleSegment2Circle, circum_1, polyList, showPolyName, showSides, ratioPolyPer2Circum_1, showVertices, inputBox1, descriptN, and sliderInstruct. A slider controls the number of sides n, and text fields display computed ratios. This resource is ideal for high school and early college students learning limits, pi approximation, and dynamic geometry, as well as for inquiry-based math labs.

GeoGebra Commands & Parameters

  1. Sequence(Rotate[A + (r, 0), ((k * 2) * π / n), A], k, 0, n) → pointList1
  2. Sequence(Segment[Element[pointList1, k], Element[pointList1, k + 1]], k, 1, n) → segList
  3. Sequence(Midpoint[Element[segList, k]], k, 1, n) → midPtList
  4. Sequence(Segment[A, Element[midPtList, k]], k, 1, n) → apothemList
  5. Element(apothemList, 1) → radius_2
  6. Circle(A, radius_2) → circle_2
  7. Circumference(circle_2) → circum_2
  8. Segment(A, A + (r, 0)) → radius_1
  9. Sequence(Rotate[radius_1, (360° * k / n), A], k, 1, n) → radiiList
  10. Circle(A, r) → circle_1
  11. Sequence(Sector[circle_1, Element[pointList1, k], Element[pointList1, k + 1]], k, 1, n) → sectorList
  12. Sequence(((Element[segList, k] * Element[apothemList, k])) / 2, k, 1, n) → apothArea
  13. Sequence(Element[sectorList, k] - Element[apothArea, k], k, 1, n) → segmentArea
  14. Sum(segmentArea, n) → valueSegAreaTotal
  15. Sum(apothArea, n) → areaPolygon
  16. Sequence(Arc[circle_1, Element[pointList1, k], Element[pointList1, k + 1]], k, 1, n) → arclist
  17. Area(circle_1) → areaCircle_1
  18. Circumference(circle_1) → circum_1
  19. If(n < 13, Element[polyList, n - 2], n + Element[polyList, 11]) → showPolyName
  20. Textfield(n) → inputBox1
  21. Area(circle_2) → areaCircle_2
  22. TableText({{"Regular " + showPolyName, TableText[{{"No.Sides", n}, {"Area", areaPolygon}, {"Perimeter", perimeterPoly}}, "c|111_"]}}, "vc_|11") → polyText
  23. TableText({{"Circle_1 A circumscribed", TableText[{{"Radius", r}, {"Area", areaCircle_1}, {"Circumference", circum_1}}, "c|111_"]}}, "vc_|11") → circle_1Text
  24. TableText({{"Shaded Space", TableText[{{"Area", valueSegAreaTotal}, {"% of Circle", (ratioCircleSegment2Circle * 100) + "\%"}}, "c|11_"]}}, "vc_|11|") → shadeText
  25. TableText({{"White Space", TableText[{{"Area", areaPolygon}, {"% of Circle", (100 * ratioPolyArea2Circle_1) + "\%"}}, "c|11_"]}}, "vc_|11|") → whiteText
  26. TableText({{"Circle_2 A circumscribed", TableText[{{"Radius", radius_2}, {"Area", areaCircle_2}, {"Circumference", circum_2}}, "c|111_"]}}, "vc_|11") → circle_2Text
  27. First(radiiList) → radius_f
  28. First(apothemList) → apothem_f
  29. TableText({{"Central Triangle A", TableText[{{"Chord", Element[segList, 1]}, {"Apothem", Element[apothemList, 1]}, {"Area", Element[apothArea, 1]}}, "c|111_"]}}, "vc_|11") → trianText

AI Drawing Prompt Reference

Draw a circle with a fixed center and adjustable radius. Inside it, generate a regular polygon whose number of sides is controlled by a slider, increasing gradually. Also draw the circumscribed and inscribed circles of the polygon, and show its sides, radii, apothems, and segments from the center to side midpoints. Use sectors and triangles to represent the area difference between the polygon and the circle, and dynamically display the ratios of the polygon's perimeter and area to those of the circle, which approach 1 as the side count increases.
The prompt above was compiled from the original drawing commands and is for reference only.