球体的体积推导与演示
2016-04-24 16:08
本GeoGebra范例通过三维构造直观演示球体体积的推导。它展示了半球与同底等高圆锥的体积之和等于同底等高圆柱的体积,从而推导出球体体积公式。该模型涉及立体几何中的球、圆锥和圆柱,帮助学习者深入理解空间几何体的体积关系及祖暅原理。
所属专题
S3数学:面积与体积
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.230.0" id="78d9cc03-bffd-483a-bebc-12a0151d630f" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="683" height="415" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.7320644216691069" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="399,118,686,655" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3,3" size="300" window="100,100,600,400" />
<view id="1" visible="false" inframe="false" stylebar="false" location="1,3" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 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="2" visible="false" inframe="false" stylebar="false" location="3,3" size="279" window="100,100,250,400" />
<view id="16" visible="true" inframe="false" stylebar="true" location="1" size="173" 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="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="true" stylebar="false" location="1" size="480" 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" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 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 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="true" inframe="false" stylebar="false" location="3" size="500" window="336,221,600,400" />
</views>
<toolbar show="false" items="0 39 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="580" height="473"/>
<coordSystem xZero="215" yZero="314.99999999999994" scale="50" yscale="49.99999999999999"/>
<evSettings axes="true" 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="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="173" height="413"/>
<coordSystem xZero="279.1" yZero="281.9" scale="55.00000000000008" yscale="54.999999999999986"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="224" g="224" b="224"/>
<axesColor r="224" g="224" b="224"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<uses3D val="true"/>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="-0.2081587829678947" yZero="0.03073155333983206" zZero="-0.8696456768680874" scale="75.15302409426519" xAngle="16" zAngle="99"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="2" show="false" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<plate show="true"/>
<bgColor r="216" g="227" b="234"/>
<clipping use="false" show="false" size="2"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="numeric" label="a">
<value val="2"/>
<show object="true" label="false" ev="3"/>
<objColor r="51" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-17" y="-78"/>
<labelMode val="1"/>
<slider min="0.01" max="2" absoluteScreenLocation="true" width="150" x="84" y="299" fixed="true" horizontal="false" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.05" speed="4" type="2" playing="false"/>
<ggbscript onUpdate="SetFilling[l_s, If[a>0.8,.5, .15 ]]"/>
</element>
<command name="If">
<input a0="a ≤ 1" a1="a" a2="If[a > 1, 1, 0]"/>
<output a0="r"/>
</command>
<element type="numeric" label="r">
<value val="1"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<labelOffset x="7" y="8"/>
</element>
<element type="numeric" label="n">
<value val="40"/>
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="16" y="-11"/>
<labelMode val="1"/>
<slider min="1" max="40" absoluteScreenLocation="true" width="100" x="41" y="503" fixed="false" horizontal="false" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="R">
<value val="1"/>
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200" x="30" y="90" fixed="false" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Sequence">
<input a0="Circle[(0, 0, (R * k / n)), sqrt(R^(2) - ((R * k / n))^(2) + ((1 - r) * ((R * k / n))^(2))), xOyPlane]" a1="k" a2="0" a3="(n * R)"/>
<output a0="c_s"/>
</command>
<element type="list" label="c_s">
<show object="false" label="true" ev="7"/>
<objColor r="0" g="102" b="153" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Cylinder[Element[c_s, j], R / n]" a1="j" a2="1" a3="(R * n)"/>
<output a0="l_s"/>
</command>
<element type="list" label="l_s">
<show object="true" label="true" ev="7"/>
<objColor r="0" g="102" b="153" alpha="0.5"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1" opacity="38"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<element type="numeric" label="u">
<value val="1"/>
<show object="false" label="true" ev="3"/>
<objColor r="255" g="255" b="255" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-14" y="79"/>
<labelMode val="1"/>
<slider min="1" max="(n * R)" absoluteScreenLocation="true" width="200" x="37" y="409" fixed="false" horizontal="false" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.05" speed="1" type="0" playing="false"/>
</element>
<command name="If">
<input a0="a > 1" a1="a - 1" a2="0"/>
<output a0="c"/>
</command>
<element type="numeric" label="c">
<value val="1"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="text1" exp=""Drag to combine the volume of the cylinder and the hemisphere.""/>
<element type="text" label="text1">
<show object="true" label="false" ev="43"/>
<objColor r="51" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1" size="0" style="1"/>
<absoluteScreenLocation x="14" y="68"/>
</element>
<expression label="M" exp="(0, 0, a)" type="point" />
<element type="point" label="M">
<show object="false" label="true" ev="7"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="2" w="1"/>
<pointSize val="3"/>
</element>
<expression label="N" exp="(0, 0, a + 1)" type="point" />
<element type="point" label="N">
<show object="false" label="true" ev="7"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="3" w="1"/>
<pointSize val="3"/>
</element>
<command name="Cone">
<input a0="N" a1="M" a2="1"/>
<output a0="b" a1="d" a2="e"/>
</command>
<element type="quadric" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="102" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="conic" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="102" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" 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="-1" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
</element>
<element type="quadric" label="e">
<show object="true" label="true" ev="4"/>
<objColor r="204" g="102" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="0" x1="0" y1="-1" z1="0" x2="0" y2="0" z2="-1"/>
</element>
</construction>
</geogebra>