球体的表面积与体积关系探索
2014-10-05 15:12
本GeoGebra数学范例直观演示了球体表面积与体积之间的内在关系。通过运用球体、正交平面、旋转、平移和镜像等三维几何构造命令，动态展示球体的几何性质。该交互式模型有助于学习者深入理解立体几何中球体的度量特征，探索其表面积和体积公式的几何意义。
所属专题
S3数学：面积与体积
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<auxiliary val="true"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[l3', ((t * 7.5))°, l0'], ((t * 15))°, l1'']" a1="((t * 15))°" a2="l2''"/>
<output a0="l3''"/>
</command>
<element type="line3d" label="l3''">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.7071067811865475" oy="0" oz="0.2928932188134524" ow="1" vx="-0.024094079294328167" vy="0.18301270189221933" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[Element[list3', 4], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2'']" a1="(15° * t)" a2="l3''"/>
<output a0="g"/>
</command>
<element type="polyhedron" label="g">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[l4', ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2'']" a1="((t * 15))°" a2="l3''"/>
<output a0="l4''"/>
</command>
<element type="line3d" label="l4''">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.8660254037844386" oy="0" oz="0.4999999999999999" ow="1" vx="-0.02950910004663077" vy="0.22414386804201344" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[Rotate[Element[list3', 5], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3'']" a1="(15° * t)" a2="l4''"/>
<output a0="h"/>
</command>
<element type="polyhedron" label="h">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Translate">
<input a0="l5" a1="Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]"/>
<output a0="l5'"/>
</command>
<element type="line3d" label="l5'">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.9659258262890683" oy="0" oz="0.7411809548974793" ow="1" vx="-0.0329131243968489" vy="0.24999999999999997" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[Rotate[l5', ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], ((t * 15))°, l3'']" a1="(15° * t)" a2="l4''"/>
<output a0="l5''"/>
</command>
<element type="line3d" label="l5''">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.9659258262890683" oy="0" oz="0.7411809548974793" ow="1" vx="-0.03291312439684892" vy="0.25" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Rotate">
<input a0="Element[list2', 1]" a1="((t * 7.5))°" a2="l0'"/>
<output a0="poly1"/>
</command>
<element type="polygon3d" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[Rotate[Rotate[Element[list3', 6], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3''], (15° * t), l4'']" a1="(15° * t)" a2="l5''"/>
<output a0="i"/>
</command>
<element type="polyhedron" label="i">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Rotate">
<input a0="Rotate[Element[list2', 2], ((t * 7.5))°, l0']" a1="((t * 15))°" a2="l1''"/>
<output a0="poly2"/>
</command>
<element type="polygon3d" label="poly2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Element[list2', 3], ((t * 7.5))°, l0'], ((t * 15))°, l1'']" a1="((t * 15))°" a2="l2''"/>
<output a0="poly3"/>
</command>
<element type="polygon3d" label="poly3">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[Element[list2', 4], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2'']" a1="(15° * t)" a2="l3''"/>
<output a0="poly4"/>
</command>
<element type="polygon3d" label="poly4">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[Rotate[Element[list2', 5], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3'']" a1="(15° * t)" a2="l4''"/>
<output a0="poly5"/>
</command>
<element type="polygon3d" label="poly5">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[Rotate[Rotate[Element[list2', 6], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3''], (15° * t), l4'']" a1="(15° * t)" a2="l5''"/>
<output a0="poly6"/>
</command>
<element type="polygon3d" label="poly6">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="OrthogonalPlane">
<input a0="(0, 0, 1)" a1="zAxis"/>
<output a0="m"/>
</command>
<element type="plane3d" label="m">
<show object="false" label="false" ev="4"/>
<objColor r="99" g="219" b="219" alpha="0.5"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0" y="0" z="1" w="-1"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="Translate">
<input a0="l6" a1="Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]"/>
<output a0="l6'"/>
</command>
<element type="line3d" label="l6'">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="1" oy="0" oz="0.9999999999999999" ow="1" vx="-0.03407417371093169" vy="0.25881904510252074" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[Rotate[Rotate[l6', ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], ((t * 15))°, l3''], (15° * t), l4'']" a1="(15° * t)" a2="l5''"/>
<output a0="l6''"/>
</command>
<element type="line3d" label="l6''">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="1" oy="0" oz="0.9999999999999999" ow="1" vx="-0.03407417371093171" vy="0.2588190451025209" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Rotate">
<input a0="Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[m, ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3''], (15° * t), l4''], (15° * t), l5'']" a1="(7.5° * t)" a2="l6''"/>
<output a0="n"/>
</command>
<element type="plane3d" label="n">
<show object="false" label="false" ev="4"/>
<objColor r="99" g="219" b="219" alpha="0.5"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0" y="0" z="1" w="-1"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="Mirror">
<input a0="i" a1="n"/>
<output a0="i'"/>
</command>
<element type="polyhedron" label="i'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="h" a1="n"/>
<output a0="h'"/>
</command>
<element type="polyhedron" label="h'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="g" a1="n"/>
<output a0="g'"/>
</command>
<element type="polyhedron" label="g'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="f" a1="n"/>
<output a0="f'"/>
</command>
<element type="polyhedron" label="f'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="e" a1="n"/>
<output a0="e'"/>
</command>
<element type="polyhedron" label="e'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="d" a1="n"/>
<output a0="d'"/>
</command>
<element type="polyhedron" label="d'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="poly6" a1="n"/>
<output a0="poly6'"/>
</command>
<element type="polygon3d" label="poly6'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="poly5" a1="n"/>
<output a0="poly5'"/>
</command>
<element type="polygon3d" label="poly5'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="poly4" a1="n"/>
<output a0="poly4'"/>
</command>
<element type="polygon3d" label="poly4'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="poly3" a1="n"/>
<output a0="poly3'"/>
</command>
<element type="polygon3d" label="poly3'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="poly2" a1="n"/>
<output a0="poly2'"/>
</command>
<element type="polygon3d" label="poly2'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Mirror">
<input a0="poly1" a1="n"/>
<output a0="poly1'"/>
</command>
<element type="polygon3d" label="poly1'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
</element>
<command name="Sequence">
<input a0="Rotate[list3, 15° + ((15° * 22) * i), zAxis]" a1="i" a2="0" a3="1"/>
<output a0="list4"/>
</command>
<element type="list" label="list4">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="Mirror">
<input a0="list4" a1="m"/>
<output a0="list4'"/>
</command>
<element type="list" label="list4'">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Rotate[list2, (15° * i), zAxis]" a1="i" a2="1" a3="23"/>
<output a0="list5"/>
</command>
<element type="list" label="list5">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="Mirror">
<input a0="list5" a1="m"/>
<output a0="list5'"/>
</command>
<element type="list" label="list5'">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
</construction>
</geogebra>