Exploring the Surface Area and Volume of Spheres
2014-10-05 15:12
This GeoGebra mathematics applet intuitively demonstrates the intrinsic relationship between the surface area and volume of a sphere. By utilizing 3D geometric construction commands such as Sphere, OrthogonalPlane, Rotate, Translate, and Mirror, it dynamically displays the geometric properties of spheres. This interactive model helps learners deeply understand the metric characteristics of spheres
In collections
S3 Area and Volume
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<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>