Geometric Illustration and Derivation of Sphere Volume
2021-09-30 15:43
This GeoGebra applet visually demonstrates the derivation and calculation of the volume of a sphere using solid geometry constructions. By employing geometric objects such as cylinders, spheres, cross-sections, and orthogonal lines, along with vector and surface commands, it illustrates the volumetric relationship between a sphere and related solids (e.g., Archimedes' theorem or Cavalieri's princi
In collections
Volume and Surface Area of Cones and Spheres
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<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"/>
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<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="StartAnimation(b,o)
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<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>