圆面积公式的几何推导与展开图演示
2020-11-04 07:45
本范例展示了圆面积公式的几何推导过程。通过将圆分割成多个扇形并重新排列，直观地演示了圆面积如何展开并近似为一个矩形，从而帮助学生深刻理解圆面积公式的几何意义。利用动态交互功能，用户可以观察分割份数增加时，展开图逐渐逼近矩形的极限过程，是理解微积分基本思想的优秀教学材料。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.613.0" app="classic" platform="w" id="9ccd4eea-0a26-45f5-86d2-0edc5c70f024" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="690" height="639" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.39565217391304347" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="true" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="395" window="853,171,701,917" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 , 538 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="true" location="1,1" size="1168" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="273" tab="ALGEBRA" window="100,100,600,400" />
<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" toolbar="0" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="1" visible="true" inframe="false" stylebar="true" location="" size="637" window="100,100,600,400" />
</views>
<toolbar show="false" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 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 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 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="3"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="688" height="636"/>
<coordSystem xZero="-15.520495656015951" yZero="767.4697782085038" scale="150.21029443520433" yscale="150.21029443520487"/>
<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="11" grid="0"/>
<labelStyle axes="1" serif="true"/>
<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" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="O">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelOffset x="0" y="3"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="1.341589112874984" y="2.8324941363591805" z="1"/>
<pointSize val="5"/>
<pointStyle val="1"/>
</element>
<element type="numeric" label="n">
<value val="28"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-11" y="-8"/>
<labelMode val="1"/>
<slider min="11" max="50" absoluteScreenLocation="true" width="200" x="77" y="56" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<caption val="分割条数"/>
</element>
<expression label="half" exp="ceil(n / 2)" />
<element type="numeric" label="half">
<value val="14"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="r" exp="1"/>
<element type="numeric" label="r">
<value val="1"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<caption val="radius"/>
</element>
<expression label="B" exp="O + (0, (-r))" type="point" />
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-10" y="28"/>
<labelMode val="0"/>
<coords x="1.341589112874984" y="1.8324941363591805" z="1"/>
<pointSize val="2"/>
<pointStyle val="6"/>
</element>
<command name="ToComplex">
<input a0="Vector[O, B]"/>
<output a0="z_{1}"/>
</command>
<element type="vector" label="z_{1}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="-1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="complex"/>
<startPoint exp="O"/>
</element>
<command name="Sequence">
<input a0="O + (z_{1} * ℯ^((((ί * k) * 2) * π / n)))" a1="k" a2="0" a3="n"/>
<output a0="Dlist"/>
</command>
<element type="list" label="Dlist">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="numeric" label="t">
<value val="0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="n + 2" absoluteScreenLocation="true" width="200" x="82" y="124" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<command name="Circle">
<input a0="O" a1="r"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="8.822884380293825" A3="0" A4="-1.341589112874984" A5="-2.8324941363591805"/>
<eqnStyle style="specific"/>
</element>
<command name="Sequence">
<input a0="CircleSector[O, Dlist(k), Dlist(k + 1)]" a1="k" a2="1" a3="half"/>
<output a0="sec1"/>
</command>
<element type="list" label="sec1">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<expression label="text1" exp=""1. 旋转用复数乘法完成 $O+z_0\times e^{i*2k\pi/n},k\in[0,n-1]$, 得到复数序列 l1
2. 扇形用序列，为了区分前后两组，可以分开：
 sec1=Sequence(CircularSector(O,l1(k),l1(k+1)),k,0,ceil(n/2))
 sec2=Sequence(CircularSector(O,l1(k),l1(k+1)),k,ceil(n/2)+1,n-1)
 这里的复数序列，自动转换成了点序列，无须进行转换操作。
 ""/>
<element type="text" label="text1">
<show object="false" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="158" y="132"/>
</element>
<expression label="θ" exp="(2 * π / n)" />
<element type="numeric" label="θ">
<value val="0.2243994752564138"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Distance">
<input a0="Dlist(1)" a1="Dlist(2)"/>
<output a0="xSeg"/>
</command>
<element type="numeric" label="xSeg">
<value val="0.22392895220661566"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<caption val="分割的弦长"/>
</element>
<command name="Sequence">
<input a0="B + ((k * xSeg), 0)" a1="k" a2="0" a3="n"/>
<output a0="Plist"/>
</command>
<element type="list" label="Plist">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="point" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2.1859701141577794" y="0.04849242529361655" z="1"/>
<pointSize val="5"/>
<pointStyle val="10"/>
</element>
<command name="Segment">
<input a0="A" a1="A + ((2 * r), 0)"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="2" z="-0.0969848505872331"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
<input a0="f"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="false" label="false" ev="8"/>
<objColor r="153" g="204" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="3.1859701141577794" y="0.04849242529361655" z="1"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="34" y="-8"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2.1859701141577794" y="0.04849242529361655" z="1"/>
<pointSize val="3"/>
<pointStyle val="2"/>
</element>
<command name="Sequence">
<input a0="Rotate[sec1(k), ((-Min[t, 1]) * Angle[Vector[Dlist(k), Dlist(k) + (1, 0)], Vector[Dlist(k), Dlist(k + 1)]]), Dlist(k)]" a1="k" a2="1" a3="half"/>
<output a0="rsec1"/>
</command>
<element type="list" label="rsec1">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="0"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Translate[rsec1(k), Vector[((Min[t - k, 1] * (t ≥ k)) * Vector[Dlist(k), Plist(k)])]]" a1="k" a2="1" a3="half"/>
<output a0="tsec1"/>
</command>
<element type="list" label="tsec1">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="CircleSector[O, Dlist(k), Dlist(k + 1)]" a1="k" a2="half + 1" a3="n"/>
<output a0="sec2"/>
</command>
<element type="list" label="sec2">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Rotate[sec2(k - half), ((-Min[t, 1]) * Angle[Vector[Dlist(k), Dlist(k) + (-1, 0)], Vector[Dlist(k), Dlist(k + 1)]]), Dlist(k)]" a1="k" a2="half + 1" a3="n"/>
<output a0="rsec2"/>
</command>
<element type="list" label="rsec2">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="UnitVector">
<input a0="Vector[Plist(1), Plist(2)]"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="1" y="0" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<expression label="O1" exp="Dlist(1) + ((r * u) * ℯ^((ί * (π - θ) / 2)))" type="point" />
<element type="point" label="O1">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="1.4535535889782918" y="2.826206346252423" z="1"/>
<coordStyle style="complex"/>
<pointSize val="2"/>
<pointStyle val="6"/>
<caption val="小扇形对应的圆心坐标"/>
</element>
<command name="Sequence">
<input a0="O1 + (k * xSeg)" a1="k" a2="0" a3="half"/>
<output a0="Olist"/>
</command>
<element type="list" label="Olist">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Translate[rsec2(k - half), Vector[((Min[t - k, 1] * (t ≥ k)) * Vector[Dlist(k), Olist(k - half + 1)])]]" a1="k" a2="half + 1" a3="n"/>
<output a0="tsec2"/>
</command>
<element type="list" label="tsec2">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<expression label="text2" exp=""圆面积展开示意图
1. 分割成全等的扇形
2. 先旋转后平移
3. 调高扇形数目$n$
4. 得出圆面积公式
$\approx$平行四边形面积
逼近长方形面积
5. 细分到无穷时，长方形的长
为半圆长$\pi r$，宽为半径$r$
6. 故圆面积=$\pi r^2$""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="208" g="240" b="192" alpha="255"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="2.7553646278097665" y="4.7437540148540585" z="1"/>
</element>
</construction>
</geogebra>