球体体积的几何图解与推导演示
2021-09-30 15:43
本 GeoGebra 范例通过立体几何构造，直观演示球体体积的推导与计算过程。利用圆柱、球体、截面及正交线等几何对象，结合向量与曲面命令，展示球体与相关几何体（如圆柱、圆锥）之间的体积关系（如祖暅原理或阿基米德球体体积定理）。适合高中或大学微积分与立体几何教学，帮助学生通过三维可视化深入理解空间几何体的体积公式及其积分思想。
所属专题
锥体与球体的体积和表面积
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<layer val="0"/>
<labelMode val="3"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="r"/>
</element>
<command name="Point">
<input a0="f_1"/>
<output a0="L"/>
</command>
<element type="point3d" label="L">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="-8.210836034688002" y="-3.4169390197432357" z="0" w="1"/>
<pointSize val="1"/>
</element>
<command name="Segment">
<input a0="F" a1="L"/>
<output a0="n"/>
</command>
<element type="segment3d" label="n">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-15" y="2"/>
<labelMode val="3"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="h"/>
</element>
<expression label="text2" exp=""\ovalbox{r^2=R^2-h^2}""/>
<element type="text" label="text2">
<show object="true" label="false" ev="37"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<absoluteScreenLocation x="465" y="668"/>
</element>
<element type="boolean" label="o">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="21" y="175"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="StartAnimation(b,o)
If(o==false,SetValue(q,false))"/>
<caption val="顯示所有截面"/>
</element>
<expression label="text1" exp=""球體體積
\\=圓柱體積-錐體體積
\\= \pi R^2\times2R-2\times\frac{1}{3}\times\pi R^2\times R
\\=\frac{4}{3}\pi R^3""/>
<element type="text" label="text1">
<show object="true" label="false" ev="43"/>
<condition showObject="o"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="55" y="235"/>
</element>
<expression label="text4" exp=""試以同樣的原理,推測球冠的體積。""/>
<element type="text" label="text4">
<show object="true" label="false" ev="43"/>
<condition showObject="o"/>
<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="54" y="462"/>
</element>
<element type="boolean" label="q">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="o"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="315" y="504"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="顯示答案"/>
</element>
<element type="button" label="button1">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="467" y="179"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetActiveView(-1)
ZoomIn(1)"/>
<font serif="false" sizeM="2" size="24" style="0"/>
<caption val="清除軌跡"/>
</element>
<command name="Surface">
<input a0="(-6, -6, 0) + (rr * (cos(t), sin(t), 0))" a1="rr" a2="s" a3="l" a4="t" a5="0" a6="(2 * π)"/>
<output a0="p"/>
</command>
<element type="surfacecartesian3d" label="p">
<show object="true" label="false" ev="7"/>
<objColor r="255" g="0" b="0" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(-6, -6, b) + (rr * (cos(t), sin(t), 0))" a1="rr" a2="s" a3="l" a4="t" a5="0" a6="(2 * π)"/>
<output a0="v"/>
</command>
<element type="surfacecartesian3d" label="v">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
<trace val="true"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
</construction>
</geogebra>