Approximating Pi Using Polygon Areas

2014-02-11 01:16

This applet demonstrates approximating Pi using areas of inscribed and circumscribed regular polygons. By adjusting the side number n, users can observe how polygon areas gradually squeeze the circle's true area as n increases. It also explores a historical question: when the Bible approximated Pi as 3, what regular polygon was it treating the circle as? Ideal for teaching limits and geometry.

#circle #polygon #limit #sequence

In collections 微積分

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.4" version="4.4.10.0" id="120ec54b-70ed-48b7-ba05-97f605616c95"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="786" height="622" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.2544529262086514" orientation="1" />
	</panes>
	<views>
		<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="740,350,700,550" />
		<view id="4" toolbar="0 59 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
		<view id="8" toolbar="1001 | 1002 | 1003  || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
		<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="775" window="100,100,600,400" />
		<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
		<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
		<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
	</views>
	<toolbar show="true" items="0 39 59 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
	<input show="true" cmd="true" top="false" />
	<dockBar show="true" east="true" />
</perspective>
	</perspectives>
	<labelingStyle  val="3"/>
	<font  size="12"/>
	<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
	<size  width="775" height="485"/>
	<coordSystem xZero="242.82470756729995" yZero="216.05692105110015" scale="129.68712300500007" yscale="129.68712300500016"/>
	<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
	<bgColor r="255" g="255" b="255"/>
	<axesColor r="0" g="0" b="0"/>
	<gridColor r="192" g="192" b="192"/>
	<lineStyle axes="1" grid="10"/>
	<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
	<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
	<continuous val="false"/>
	<usePathAndRegionParameters val="true"/>
	<decimals val="2"/>
	<angleUnit val="degree"/>
	<algebraStyle val="0"/>
	<coordStyle val="0"/>
	<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="n">
	<value val="8.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="3" max="20" absoluteScreenLocation="true" width="300.0" x="96.0" y="445.0" fixed="false" horizontal="true"/>
	<lineStyle thickness="4" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="point" label="O">
	<show object="false" label="true"/>
	<objColor r="0" g="0" b="255" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="1" playing="false"/>
	<coords x="0.0" y="0.0" z="1.0"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Sequence">
	<input a0="(1; π / 2 + ((2 * π / n) * k))" a1="k" a2="1" a3="n"/>
	<output a0="v"/>
</command>
<element type="list" label="v">
	<show object="false" label="false"/>
	<objColor r="0" g="100" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<pointSize val="3"/>
	<pointStyle val="-1"/>
</element>
<command name="Sequence">
	<input a0="Segment[O, Element[v, k]]" a1="k" a2="1" a3="n"/>
	<output a0="r"/>
</command>
<element type="list" label="r">
	<show object="true" label="false"/>
	<objColor r="96" g="96" b="96" alpha="0.0"/>
	<layer val="3"/>
	<labelMode val="0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<pointSize val="3"/>
	<pointStyle val="-1"/>
</element>
<command name="Polygon">
	<input a0="v"/>
	<output a0="p" a1="" a2="" a3="" a4="" a5="" a6="" a7="" a8=""/>
</command>
<element type="polygon" label="p">
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<show object="true" label="false"/>
	<objColor r="51" g="51" b="255" alpha="0.5"/>
	<layer val="3"/>
	<labelMode val="0"/>
</element>
<command name="Circle">
	<input a0="O" a1="1"/>
	<output a0="c"/>
</command>
<element type="conic" label="c">
	<show object="true" label="false"/>
	<objColor r="255" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelMode val="0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<eigenvectors  x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
	<matrix A0="1.0" A1="1.0" A2="-1.0" A3="0.0" A4="-0.0" A5="-0.0"/>
	<eqnStyle style="specific"/>
</element>
<expression label="R" exp="sec(π / n)" />
<element type="numeric" label="R">
	<value val="1.082392200292394"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sequence">
	<input a0="(R; π / 2 + ((2 * π / n) * k))" a1="k" a2="1" a3="n"/>
	<output a0="v2"/>
</command>
<element type="list" label="v2">
	<show object="false" label="false"/>
	<objColor r="0" g="100" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<pointSize val="3"/>
	<pointStyle val="-1"/>
</element>
<command name="Polygon">
	<input a0="v2"/>
	<output a0="p2" a1="" a2="" a3="" a4="" a5="" a6="" a7="" a8=""/>
</command>
<element type="polygon" label="p2">
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<show object="true" label="false"/>
	<objColor r="51" g="51" b="255" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="0"/>
</element>
<command name="Circle">
	<input a0="O" a1="1"/>
	<output a0="c2"/>
</command>
<element type="conic" label="c2">
	<show object="true" label="false"/>
	<objColor r="255" g="255" b="255" alpha="1.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="1" type="0" typeHidden="1"/>
	<eigenvectors  x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
	<matrix A0="1.0" A1="1.0" A2="-1.0" A3="0.0" A4="-0.0" A5="-0.0"/>
	<eqnStyle style="specific"/>
</element>
<expression label="text1" exp="&quot;a_{&quot; + (LaTeX[n]) + &quot;}=&quot; + (LaTeX[p]) + &quot;&lt;\pi&lt;&quot; + (LaTeX[p2]) + &quot;=b_{&quot; + (LaTeX[n]) + &quot;}&quot;" />
<element type="text" label="text1">
	<show object="true" label="false"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2.0" size="12" style="0"/>
	<decimals val="4"/>
	<startPoint  x="0.9387616103412686" y="1.1145047997211526" z="1.0"/>
</element>
<expression label="&quot;a_&quot; + (FormulaText[n]) + &quot;=&quot; + (FormulaText[p]) + &quot;&lt;\pi&lt;&quot; + (FormulaText[p2]) + &quot;=b_&quot; + (FormulaText[n]) + &quot;&quot;" exp="&quot;a_&quot; + (LaTeX[n]) + &quot;=&quot; + (LaTeX[p]) + &quot;&lt;\pi&lt;&quot; + (LaTeX[p2]) + &quot;=b_&quot; + (LaTeX[n]) + &quot;&quot;" />
<element type="text" label="&quot;a_&quot; + (FormulaText[n]) + &quot;=&quot; + (FormulaText[p]) + &quot;&lt;\pi&lt;&quot; + (FormulaText[p2]) + &quot;=b_&quot; + (FormulaText[n]) + &quot;&quot;">
	<show object="false" label="false"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="false" sizeM="1.0" size="0" style="0"/>
</element>
<element type="button" label="button1">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelOffset x="416" y="431"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="0" playing="false"/>
	<auxiliary val="true"/>
	<ggbscript val="SetValue[n,n+1]"/>
	<font serif="false" sizeM="2.0" size="12" style="0"/>
	<file name="/geogebra/gui/images/go-next.png"/>
</element>
<element type="button" label="button2">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelOffset x="50" y="431"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="0" playing="false"/>
	<auxiliary val="true"/>
	<ggbscript val="SetValue[n,n-1]"/>
	<font serif="false" sizeM="2.0" size="12" style="0"/>
	<file name="/geogebra/gui/images/go-previous.png"/>
</element>
</construction>
</geogebra>

Loading the GeoGebra board and drawing, please wait…

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.4" version="4.4.10.0" id="120ec54b-70ed-48b7-ba05-97f605616c95"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="786" height="622" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.2544529262086514" orientation="1" />
	</panes>
	<views>
		<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="740,350,700,550" />
		<view id="4" toolbar="0 59 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
		<view id="8" toolbar="1001 | 1002 | 1003  || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
		<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="775" window="100,100,600,400" />
		<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
		<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
		<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
	</views>
	<toolbar show="true" items="0 39 59 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
	<input show="true" cmd="true" top="false" />
	<dockBar show="true" east="true" />
</perspective>
	</perspectives>
	<labelingStyle  val="3"/>
	<font  size="12"/>
	<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
	<size  width="775" height="485"/>
	<coordSystem xZero="242.82470756729995" yZero="216.05692105110015" scale="129.68712300500007" yscale="129.68712300500016"/>
	<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
	<bgColor r="255" g="255" b="255"/>
	<axesColor r="0" g="0" b="0"/>
	<gridColor r="192" g="192" b="192"/>
	<lineStyle axes="1" grid="10"/>
	<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
	<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
	<continuous val="false"/>
	<usePathAndRegionParameters val="true"/>
	<decimals val="2"/>
	<angleUnit val="degree"/>
	<algebraStyle val="0"/>
	<coordStyle val="0"/>
	<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="n">
	<value val="8.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="3" max="20" absoluteScreenLocation="true" width="300.0" x="96.0" y="445.0" fixed="false" horizontal="true"/>
	<lineStyle thickness="4" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="point" label="O">
	<show object="false" label="true"/>
	<objColor r="0" g="0" b="255" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="1" playing="false"/>
	<coords x="0.0" y="0.0" z="1.0"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Sequence">
	<input a0="(1; π / 2 + ((2 * π / n) * k))" a1="k" a2="1" a3="n"/>
	<output a0="v"/>
</command>
<element type="list" label="v">
	<show object="false" label="false"/>
	<objColor r="0" g="100" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<pointSize val="3"/>
	<pointStyle val="-1"/>
</element>
<command name="Sequence">
	<input a0="Segment[O, Element[v, k]]" a1="k" a2="1" a3="n"/>
	<output a0="r"/>
</command>
<element type="list" label="r">
	<show object="true" label="false"/>
	<objColor r="96" g="96" b="96" alpha="0.0"/>
	<layer val="3"/>
	<labelMode val="0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<pointSize val="3"/>
	<pointStyle val="-1"/>
</element>
<command name="Polygon">
	<input a0="v"/>
	<output a0="p" a1="" a2="" a3="" a4="" a5="" a6="" a7="" a8=""/>
</command>
<element type="polygon" label="p">
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<show object="true" label="false"/>
	<objColor r="51" g="51" b="255" alpha="0.5"/>
	<layer val="3"/>
	<labelMode val="0"/>
</element>
<command name="Circle">
	<input a0="O" a1="1"/>
	<output a0="c"/>
</command>
<element type="conic" label="c">
	<show object="true" label="false"/>
	<objColor r="255" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelMode val="0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<eigenvectors  x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
	<matrix A0="1.0" A1="1.0" A2="-1.0" A3="0.0" A4="-0.0" A5="-0.0"/>
	<eqnStyle style="specific"/>
</element>
<expression label="R" exp="sec(π / n)" />
<element type="numeric" label="R">
	<value val="1.082392200292394"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sequence">
	<input a0="(R; π / 2 + ((2 * π / n) * k))" a1="k" a2="1" a3="n"/>
	<output a0="v2"/>
</command>
<element type="list" label="v2">
	<show object="false" label="false"/>
	<objColor r="0" g="100" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<pointSize val="3"/>
	<pointStyle val="-1"/>
</element>
<command name="Polygon">
	<input a0="v2"/>
	<output a0="p2" a1="" a2="" a3="" a4="" a5="" a6="" a7="" a8=""/>
</command>
<element type="polygon" label="p2">
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<show object="true" label="false"/>
	<objColor r="51" g="51" b="255" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="0"/>
</element>
<command name="Circle">
	<input a0="O" a1="1"/>
	<output a0="c2"/>
</command>
<element type="conic" label="c2">
	<show object="true" label="false"/>
	<objColor r="255" g="255" b="255" alpha="1.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="1" type="0" typeHidden="1"/>
	<eigenvectors  x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
	<matrix A0="1.0" A1="1.0" A2="-1.0" A3="0.0" A4="-0.0" A5="-0.0"/>
	<eqnStyle style="specific"/>
</element>
<expression label="text1" exp="&quot;a_{&quot; + (LaTeX[n]) + &quot;}=&quot; + (LaTeX[p]) + &quot;&lt;\pi&lt;&quot; + (LaTeX[p2]) + &quot;=b_{&quot; + (LaTeX[n]) + &quot;}&quot;" />
<element type="text" label="text1">
	<show object="true" label="false"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2.0" size="12" style="0"/>
	<decimals val="4"/>
	<startPoint  x="0.9387616103412686" y="1.1145047997211526" z="1.0"/>
</element>
<expression label="&quot;a_&quot; + (FormulaText[n]) + &quot;=&quot; + (FormulaText[p]) + &quot;&lt;\pi&lt;&quot; + (FormulaText[p2]) + &quot;=b_&quot; + (FormulaText[n]) + &quot;&quot;" exp="&quot;a_&quot; + (LaTeX[n]) + &quot;=&quot; + (LaTeX[p]) + &quot;&lt;\pi&lt;&quot; + (LaTeX[p2]) + &quot;=b_&quot; + (LaTeX[n]) + &quot;&quot;" />
<element type="text" label="&quot;a_&quot; + (FormulaText[n]) + &quot;=&quot; + (FormulaText[p]) + &quot;&lt;\pi&lt;&quot; + (FormulaText[p2]) + &quot;=b_&quot; + (FormulaText[n]) + &quot;&quot;">
	<show object="false" label="false"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="false" sizeM="1.0" size="0" style="0"/>
</element>
<element type="button" label="button1">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelOffset x="416" y="431"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="0" playing="false"/>
	<auxiliary val="true"/>
	<ggbscript val="SetValue[n,n+1]"/>
	<font serif="false" sizeM="2.0" size="12" style="0"/>
	<file name="/geogebra/gui/images/go-next.png"/>
</element>
<element type="button" label="button2">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="3"/>
	<labelOffset x="50" y="431"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="0" playing="false"/>
	<auxiliary val="true"/>
	<ggbscript val="SetValue[n,n-1]"/>
	<font serif="false" sizeM="2.0" size="12" style="0"/>
	<file name="/geogebra/gui/images/go-previous.png"/>
</element>
</construction>
</geogebra>