Demonstration of Liu Hui's Circle Division Method
2014-02-06 03:48
This GeoGebra applet dynamically demonstrates the Circle Division Method by ancient Chinese mathematician Liu Hui. By continuously increasing the sides of an inscribed regular polygon, it visually illustrates how its area and perimeter progressively approximate the circle's. Utilizing Sequence and Vector commands, this construction vividly embodies the concept of limits and Riemann sums, serving a
In collections
Circumference of a Circle
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.2" version="4.2.47.0" id="31fb40a4-1c15-4e1c-9d1a-75a508e931e9" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="952" height="687" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.0" orientation="0" />
</panes>
<views>
<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="2,2" size="200" window="0,0,400,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="2" size="571" window="0,0,400,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="0" size="200" window="0,0,200,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 6" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<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="4097" visible="false" inframe="true" stylebar="true" location="1" size="150" window="390,245,500,500" />
<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="false" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="2"/>
<font size="16"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="944" height="544"/>
<coordSystem xZero="217.34741951987795" yZero="278.82288578273506" scale="34.77827261482536" yscale="34.77827261482536"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="0" rightAngleStyle="1" checkboxSize="13" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="64" g="64" b="64"/>
<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="5"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="教育局 評估及考評局組" date="">
<element type="numeric" label="o">
<value val="0.0"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="8" absoluteScreenLocation="true" width="100.0" x="87.0" y="73.0" fixed="false" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="n" exp="6 * 2^(o)" />
<element type="numeric" label="n">
<value val="6.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<element type="numeric" label="R">
<value val="5.0"/>
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<expression label="P_n" exp="(R * cos(360 * ° / n), R * sin(360 * ° / n))" type="point" />
<element type="point" label="P_n">
<show object="false" label="false"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.5000000000000004" y="4.330127018922193" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Distance">
<input a0="(5, 0)" a1="P_n"/>
<output a0="l"/>
</command>
<element type="numeric" label="l">
<value val="4.999999999999999"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="O" exp="(R - R, R - R)" type="point" />
<element type="point" label="O">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="boolean" label="g">
<value val="false"/>
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="454" y="196"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="q">
<value val="false"/>
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="454" y="243"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="r">
<value val="false"/>
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="454" y="297"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="point" label="F">
<show object="false" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="6.804609967294387" y="4.624234434063845" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="F" a1="xAxis"/>
<output a0="s"/>
</command>
<element type="line" label="s">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="0.0" z="6.804609967294387"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<expression label="T11" exp=""放大／縮小：滑鼠滾輪向下／向上； 移動圖形：Shift+Drag""/>
<element type="text" label="T11">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.25" size="4" style="0"/>
<absoluteScreenLocation x="48" y="39"/>
</element>
<command name="Ray">
<input a0="O" a1="(0, -R)"/>
<output a0="j1"/>
</command>
<element type="ray" label="j1">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.0" y="0.0" z="-0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="j1"/>
<output a0="N"/>
</command>
<element type="point" label="N">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="-6.187113333671976" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="O" a1="N"/>
<output a0="w"/>
</command>
<element type="vector" label="w">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="-6.187113333671976" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="O"/>
</element>
<command name="Vector">
<input a0="O" a1="(R, 0)"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.0" y="0.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="O"/>
</element>
<command name="Translate">
<input a0="N" a1="u"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="false" label="false"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.0" y="-6.187113333671976" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="O" a1="(-R, 0)"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.0" y="0.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="O"/>
</element>
<command name="Translate">
<input a0="N" a1="v"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="false" label="false"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.0" y="-6.187113333671976" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="N" a1="P"/>
<output a0="z"/>
</command>
<element type="vector" label="z">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.0" y="0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="N"/>
</element>
<command name="Vector">
<input a0="N" a1="M"/>
<output a0="u1"/>
</command>
<element type="vector" label="u1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.0" y="0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="N"/>
</element>
<expression label="T1" exp=""直徑 = " + (2 * R) + "cm"" />
<element type="text" label="T1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.25" size="4" style="0"/>
<startPoint exp="N"/>
<labelOffset x="-45" y="23"/>
</element>
<expression label="T2" exp=""正" + n + "邊形"" />
<element type="text" label="T2">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.25" size="4" style="0"/>
<absoluteScreenLocation x="231" y="81"/>
</element>
<command name="Circle">
<input a0="O" a1="R"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="160" g="160" b="160" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="15" 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="-25.0" A3="0.0" A4="-0.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Sequence">
<input a0="Segment[(R * cos(360 * ° / n * i), R * sin(360 * ° / n * i)), (R * cos(360 * ° / n * (i + 1)), R * sin(360 * ° / n * (i + 1)))]" a1="i" a2="0" a3="n - 1"/>
<output a0="L1"/>
</command>
<element type="list" label="L1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Segment[(R * cos(360 * ° / n * i), R * sin(360 * ° / n * i)), O]" a1="i" a2="0" a3="n - 1"/>
<output a0="L2"/>
</command>
<element type="list" label="L2">
<show object="true" label="false"/>
<objColor r="192" g="192" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Segment">
<input a0="P" a1="(-R, 0)"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-6.187113333671976" y="0.0" z="-30.935566668359883"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="(R, 0)" a1="M"/>
<output a0="b"/>
</command>
<element type="segment" label="b">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.187113333671976" y="0.0" z="-30.935566668359883"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="T8" exp=""邊長 = " + l + " cm"" />
<element type="text" label="T8">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.25" size="4" style="0"/>
<absoluteScreenLocation x="481" y="213"/>
</element>
<expression label="T9" exp=""周界 = " + l + " × " + n + " = " + (l * n) + " cm"" />
<element type="text" label="T9">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.25" size="4" style="0"/>
<absoluteScreenLocation x="481" y="254"/>
</element>
<expression label="T4" exp=""\frac{周界}{直徑}=\frac{" + (LaTeX[l * n]) + " \; cm}{" + (LaTeX[2 * R]) + " \; cm}=" + (LaTeX[l * n / (2 * R)]) + """ />
<element type="text" label="T4">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.375" size="6" style="0"/>
<absoluteScreenLocation x="477" y="284"/>
</element>
</construction>
</geogebra>