展开圆面积动态演示(讲义版)
2014-03-15 04:03
本GeoGebra范例通过动态几何构造,直观演示了圆面积公式的推导过程。利用序列(Sequence)等命令将圆形分割并展开,帮助学生理解圆面积与半径及周长的关系。适用于课堂教学辅助,结合滑块动画展示极限思想在几何面积计算中的应用。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.4" version="4.4.14.0" id="01ed62d0-507d-49da-9eb1-0d08d6832aa7" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="872" height="599" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.22935779816513763" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="787,334,923,550" />
<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="652" window="100,100,600,400" />
<view id="2" visible="true" 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="265" 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"/>
<consProtColumns col0="true" col1="true" col2="false" col3="true" col4="false" col5="true" col6="true" col7="false"/>
<consProtocol useColors="true" addIcons="false" showOnlyBreakpoints="false"/>
</gui>
<euclidianView>
<size width="652" height="443"/>
<coordSystem xZero="132.6507545973693" yZero="401.9741619159399" scale="38.9072631172055" yscale="38.90726311720553"/>
<evSettings axes="false" grid="true" 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="t">
<value val="0.8"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="3" absoluteScreenLocation="true" width="200.0" x="16.0" y="33.0" fixed="false" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="0.05" speed="2" type="0" playing="true"/>
</element>
<expression label ="f" exp="f(t, a, b) = (abs(t - a) - abs(t - b) + abs(a - b)) / 2 + a"/>
<element type="functionNVar" label="f">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="m">
<value val="10.0"/>
<show object="true" label="true"/>
<objColor r="127" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="2" max="20" absoluteScreenLocation="true" width="200.0" x="15.0" y="83.0" 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="(2 * m)" />
<element type="numeric" label="n">
<value val="20.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<element type="numeric" label="r">
<value val="2.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="θ" exp="π / n" />
<element type="numeric" label="θ">
<value val="0.15707963267948966"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="α" exp="(2 * θ)" />
<element type="numeric" label="α">
<value val="0.3141592653589793"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="h" exp="(r * cos(θ))" />
<element type="numeric" label="h">
<value val="1.9753766811902755"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="w" exp="((2 * r) * sin(θ))" />
<element type="numeric" label="w">
<value val="0.6257378601609235"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="t1" exp="f(1 - t, 0, 1)" />
<element type="numeric" label="t1">
<value val="0.19999999999999996"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="z" exp="cos((t1 * α)) + (ί * sin((t1 * α)))" type="point" />
<element type="point" label="z">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.9980267284282716" y="0.06279051952931336" z="1.0"/>
<coordStyle style="complex"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="β" exp="(t1 * α)" />
<element type="numeric" label="β">
<value val="0.06283185307179585"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sequence">
<input a0="z^(k)" a1="k" a2="1" a3="n"/>
<output a0="zk"/>
</command>
<element type="list" label="zk">
<show object="false" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="ρ" exp="cos(π / 2 - θ) + (ί * sin(π / 2 - θ))" type="point" />
<element type="point" label="ρ">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.15643446504023092" y="0.9876883405951378" z="1.0"/>
<coordStyle style="complex"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Join">
<input a0="{{0 + 0ί}, Sequence[(w * Sum[zk, k]), k, 1, n]}"/>
<output a0="sk"/>
</command>
<element type="list" label="sk">
<show object="false" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="2"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="((1 - (r * ρ / w)) * Element[sk, k]) + ((r * ρ / w) * Element[sk, k + 1])" a1="k" a2="1" a3="n"/>
<output a0="vk"/>
</command>
<element type="list" label="vk">
<show object="false" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="CircleSector[Element[vk, k], Element[sk, k], Element[sk, k + 1]]" a1="k" a2="1" a3="m"/>
<output a0="pie"/>
</command>
<element type="list" label="pie">
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.16"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<expression label="c" exp="(Element[vk, m] + (3 * Element[vk, m + 1])) / 4" type="point" />
<element type="point" label="c">
<show object="false" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-16" y="-4"/>
<labelMode val="0"/>
<coords x="4.674916543232062" y="3.7299262832222997" z="1.0"/>
<coordStyle style="complex"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<expression label="t2" exp="f(t - 1, 0, 1)" />
<element type="numeric" label="t2">
<value val="0.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="t3" exp="f(t - 2, 0, 1)" />
<element type="numeric" label="t3">
<value val="-1.1102230246251565E-16"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="v" exp="((-h) * ί)" type="point" />
<element type="point" label="v">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.0" y="-1.9753766811902755" z="1.0"/>
<coordStyle style="complex"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="CircleSector[Rotate[Element[vk, k], (t2 * π), c] + (t3 * v), Rotate[Element[sk, k], (t2 * π), c] + (t3 * v), Rotate[Element[sk, k + 1], (t2 * π), c] + (t3 * v)]" a1="k" a2="m + 1" a3="n"/>
<output a0="pie2"/>
</command>
<element type="list" label="pie2">
<show object="true" label="false"/>
<objColor r="127" g="0" b="255" alpha="0.52"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
</construction>
</geogebra>