利用正多边形面积逼近圆周率
2014-02-11 01:16
本范例演示了利用圆内接与外切正多边形的面积来逼近圆周率π的数学过程。通过动态调整正多边形的边数n,用户可以直观观察到内接正多边形面积与外切正多边形面积如何随着n的增大而逐渐夹逼圆的真实面积,从而估算π的近似值。此外,本范例还结合数学史,探讨了《圣经》中将圆周率近似为3时,实际上是将圆视作多少边的正多边形。适合用于极限思想、几何面积计算及数学史的教学演示。
所属专题
微积分
GeoGebra 画板及绘图内容加载中,请稍等……
<?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=""a_{" + (LaTeX[n]) + "}=" + (LaTeX[p]) + "<\pi<" + (LaTeX[p2]) + "=b_{" + (LaTeX[n]) + "}"" />
<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=""a_" + (FormulaText[n]) + "=" + (FormulaText[p]) + "<\pi<" + (FormulaText[p2]) + "=b_" + (FormulaText[n]) + """ exp=""a_" + (LaTeX[n]) + "=" + (LaTeX[p]) + "<\pi<" + (LaTeX[p2]) + "=b_" + (LaTeX[n]) + """ />
<element type="text" label=""a_" + (FormulaText[n]) + "=" + (FormulaText[p]) + "<\pi<" + (FormulaText[p2]) + "=b_" + (FormulaText[n]) + """>
<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>