Polygon Limit Exploration
Author: A B Cron

Looking at LTF Polygons in a Circle, this applet will assist in completing the exercise.

The circle on the screen is a unit circle; however, you can change the radius to match any value you measure up to 8 units. The table will give the results up to a 100-gon.

Table format from LTF activity on Limits of a Polygon in a Circle.

Triangle ABC is triangle formed by the point A and any side of the polygon.  The area of the polygon is the number of sides times the area of Triangle ABC.

If you expand the radius of the circle and can no longer see the figure, click mouse on point A and scroll the figure in and out...

Excellent Resource: http://www.mathopenref.com/polygonregular.html

Polomery

Apotém

Stredy strán

Kruhové výseky

Kruhové odseky

Strany n-uholníka

\textcolor{red}{Mnohouholník}  \; \textcolor{black}{a}  \; \textcolor{green}{Kruh}

Vrcholy n-uholníka

n = :

počet strán

Move slider to change radius
of the circle at the left.

Kružnica opísaná

Kružnica vpísaná

Reset

\textcolor{blue}{Archimedes} 

S = (1, 1)
r = 1 (slider from 0.1 to 10, step 0.1)
n = 96
pointList1 = Sequence(Rotate[S + (r, 0), ((k * 2) * pi / n), S], k, 0, n)
segList = Sequence(Segment[Element[pointList1, k], Element[pointList1, k + 1]], k, 1, n)
perimeterPoly: (Element[segList, 1] * n)
midPtList = Sequence(Midpoint[Element[segList, k]], k, 1, n)
apothemList = Sequence(Segment[S, Element[midPtList, k]], k, 1, n)
radius_2 = Element(apothemList, 1)
circle_2 = Circle(S, radius_2)
circum_2 = Circumference(circle_2)
radius_1 = Segment(S, S + (r, 0))
radiiList = Sequence(Rotate[radius_1, (360° * k / n), S], k, 1, n)
circle_1 = Circle(S, r)
showRadii = false
showApothem = false
showMidPoints = false
sectorList = Sequence(Sector[circle_1, Element[pointList1, k], Element[pointList1, k + 1]], k, 1, n)
apothArea = Sequence(((Element[segList, k] * Element[apothemList, k])) / 2, k, 1, n)
segmentArea = Sequence(Element[sectorList, k] - Element[apothArea, k], k, 1, n)
showSector = false
valueSegAreaTotal = Sum(segmentArea, n)
areaPolygon = Sum(apothArea, n)
areaPolyText: "Area of polygon = " + areaPolygon + ""
arclist = Sequence(Arc[circle_1, Element[pointList1, k], Element[pointList1, k + 1]], k, 1, n)
areaCircle_1 = Area(circle_1)
sumSegCirText: "Sum Segments of Circle = " + valueSegAreaTotal + ""
showSegment = true
ratioPolyArea2Circle_1: areaPolygon / areaCircle_1
ratioCircleSegment2Circle: valueSegAreaTotal / areaCircle_1
circum_1 = Circumference(circle_1)
polyList: {"Triangle", "Square", "Pentagon", "Hexagon", "Heptagon", "Octagon", "Nonagon", "Decagon", "-gon"}
showPolyName = If(n < 11, Element[polyList, n - 2], n + Element[polyList, 9])
showSides = true
ratioPolyPer2Circum_1: perimeterPoly / circum_1
showVertices = false
inputBox1 = Textfield(n)
areaCircle_2 = Area(circle_2)
ratioCircum_22PolyPer: circum_2 / perimeterPoly
ratioAreaCircle_22PolyArea: areaCircle_2 / areaPolygon
poly2cir_2: "$ \frac{Area\; Circle_2}{Area\; Polygon} =  " + (LaTeX[(ratioAreaCircle_22PolyArea * 100)]) + "  \;\% $"
ratioAround_2: "$ \frac{Circumference_2}{Polygon \;Perimeter} = " + (LaTeX[(ratioCircum_22PolyPer * 100)]) + "  \;\% $"
showCircle_1 = true
showCircle_2 = false
areaCircleText: "Oblasť kruhu A= " + areaCircle_1 + ""
circle_2Text = TableText({{"Circle_2 A ohraničené", TableText[{{"Radius", radius_2}, {"Area", areaCircle_2}, {"Circumference", circum_2}}, "c|111_"]}}, "vc_|11")
polyText = TableText({{"Pravidelný " + showPolyName, TableText[{{"Poč. strán", n}, {"Obsah", areaPolygon}, {"Obvod", perimeterPoly}}, "c|111_"]}}, "vc_|11")
trianText = TableText({{"\triangle ABS - stredový", TableText[{{"Tetiva", Element[segList, 1]}, {"Výška", Element[apothemList, 1]}, {"Obsah", Element[apothArea, 1]}}, "c|111_"]}}, "vc_|11")
circle_1Text = TableText({{"Kruh", TableText[{{"Polomer", r}, {"Obsah", areaCircle_1}, {"Obvod", circum_1}}, "c|111_"]}}, "vc_|11")
whiteText = TableText({{"Biela plocha", TableText[{{"Obsah", areaPolygon}, {"% z Kruhu", (100 * ratioPolyArea2Circle_1) + "\%"}}, "c|11_"]}}, "vc_|11|")
shadeText = TableText({{"Šedá plocha", TableText[{{"Obsah", valueSegAreaTotal}, {"% z Kruhu", (ratioCircleSegment2Circle * 100) + "\%"}}, "c|11_"]}}, "vc_|11|")
ratioAround_1: "$ \frac{Obvod \;  n-uholníka}{Opísaná \;  kružnica} = " + (LaTeX[(ratioPolyPer2Circum_1 * 100)]) + " \;\% $"
poly2cir_1: "$ \frac{Obsah\; n-uholníka}{Obsah\; Kruhu} = " + (LaTeX[(ratioPolyArea2Circle_1 * 100)]) + "  \;\% $"
text1: "\frac{Obvod}{Priemer} = \frac{" + (LaTeX[perimeterPoly]) + "}{" + (LaTeX[(2 * r)]) + "} =" + (LaTeX[perimeterPoly / ((2 * r))]) + ""