Surface Area of a Sphere Derivation and 3D Demo
2017-11-13 02:25
This GeoGebra applet demonstrates the calculation of a sphere's surface area. By approximating the sphere with pyramids and cylinders using sequences and definite integrals, it visually illustrates the formula's derivation. Featuring 3D constructions, it helps learners understand solid geometry and calculus applications.
In collections
Volume and Surface Area of Cones and Spheres
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.398.0" app="classic" platform="d" id="38b91e1b-08f4-45bd-8e09-87dfccbeea94" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1920" height="1040" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.36875" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="597,326,700,550" />
<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="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="3" size="708" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="367" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" 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="1" size="1180" window="100,100,600,400" />
</views>
<toolbar show="true" items="0 39 73 62 | 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 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="708" height="763"/>
<coordSystem xZero="216.0" yZero="576.0" scale="49.999999999999986" yscale="50.00000000000002"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="250" b="205"/>
<axesColor r="0" g="0" b="0"/>
<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"/>
<startAnimation val="true"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.7604181421240135" yZero="-0.6903655537070064" zZero="-1.1431816322331905" scale="56.04964615432753" xAngle="20.0" zAngle="2599.067009953192"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<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="1" showNumbers="true"/>
<plate show="false"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="false" show="false" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="numeric" label="k">
<value val="18.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="n" absoluteScreenLocation="true" width="200.0" x="40.0" y="107.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="2" type="0" playing="true"/>
</element>
<element type="numeric" label="n">
<value val="20.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="5" max="20" absoluteScreenLocation="true" width="200.0" x="30.0" y="50.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Random">
<input a0="1" a1="n + k"/>
<output a0="b"/>
</command>
<element type="numeric" label="b">
<value val="3.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<coords x="6.0" y="0.0" z="1.0"/>
<coordStyle style="cartesian3d"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Rotate[A, ((-i) * π / ((2 * n))), yAxis]" a1="i" a2="0" a3="n"/>
<output a0="list_1"/>
</command>
<element type="list" label="list_1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Sequence[Polygon[Rotate[list_1(n - i + 1), ((2 * j) * π / (n + i - 1)), zAxis], Rotate[list_1(n - i), ((2 * j) * π / (n + i)), zAxis], Rotate[list_1(n - i), ((2 * (j + 1)) * π / (n + i)), zAxis]], j, 1, n + i]" a1="i" a2="0" a3="n"/>
<output a0="list_2"/>
</command>
<element type="list" label="list_2">
<show object="false" label="false" ev="4"/>
<condition showObject="h"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Pyramid">
<input a0="Element[list_2, k, b]" a1="(0, 0)"/>
<output a0="a" a1="face1" a2="face2" a3="face3" a4="edge1" a5="edge2" a6="edge3"/>
</command>
<element type="polyhedron" label="a">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="false" label="false" ev="4"/>
<condition showObject="h"/>
<objColor r="153" g="51" b="0" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
</element>
<element type="polygon3d" label="face1">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="false" label="false" ev="4"/>
<condition showObject="h"/>
<objColor r="153" g="51" b="0" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
</element>
<element type="polygon3d" label="face2">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="false" label="false" ev="4"/>
<condition showObject="h"/>
<objColor r="153" g="51" b="0" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
</element>
<element type="polygon3d" label="face3">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="false" label="false" ev="4"/>
<condition showObject="h"/>
<objColor r="153" g="51" b="0" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
</element>
<element type="segment3d" label="edge1">
<show object="false" label="false" ev="4"/>
<condition showObject="h"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edge2">
<show object="false" label="false" ev="4"/>
<condition showObject="h"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edge3">
<show object="false" label="false" ev="4"/>
<condition showObject="h"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="polygon3d" label="">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<condition showObject="h"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
</element>
<element type="polygon3d" label="">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
</element>
<expression label="text1" exp=""球體體積可視為每一小部分 \\表面(s_i)連接到圓心的體積集合, \\當每小塊面積非常小時, \\可用錐體體積逼近 \\即\Sigma\frac{1}{3}s_iR=\frac{1}{3}SR=\frac{4}{3} π R^3 \\⇒S=4 π R^2""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<condition showObject="h"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="-3.9199999999999995" y="9.339999999999996" z="1.0"/>
</element>
<command name="Mirror">
<input a0="list_2" a1="xOyPlane"/>
<output a0="list_3"/>
</command>
<element type="list" label="list_3">
<show object="false" label="false" ev="4"/>
<condition showObject="h"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Sequence[Pyramid[Element[list_2, i, j], (0, 0)], j, 1, n + i]" a1="i" a2="1" a3="10"/>
<output a0="list_4"/>
</command>
<element type="list" label="list_4">
<show object="true" label="false" ev="4"/>
<condition showObject="c"/>
<objColor r="255" g="0" b="255" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<expression label="text2" exp=""設a為球冠的高,h是球心到球冠的距離=R-a \\球冠表面形成的角錐體積=球冠體積+圓錐體積 \\\frac{1}{3}RS=\frac{ π }{3}(a^2(3R-a))+\frac{1}{3}(R^2-h^2)h π \\RS= π (3a^2R-a^3+R^2h-h^3) \\RS= π (3Ra^2-a^3+R^3-R^2a-R^3+3R^2a-3Ra^2+a^3) \\S=2 π Ra \\相當於外切圓柱等高的環面面積""/>
<element type="text" label="text2">
<show object="true" label="true" ev="40"/>
<condition showObject="c"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="-3.9599999999999995" y="2.56" z="1.0"/>
</element>
<command name="Circle">
<input a0="(0, 0, z(list_1(11)))" a1="x(A)" a2="zAxis"/>
<output a0="d"/>
</command>
<element type="conic3d" label="d">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-36.0" A3="0.0" A4="-0.0" A5="-0.0"/>
<eqnStyle style="implicit"/>
</element>
<command name="Cylinder">
<input a0="d" a1="x(A) - z(list_1(11))"/>
<output a0="e" a1="f" a2="g"/>
</command>
<element type="quadriclimited" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
</element>
<element type="conic3d" label="f">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-36.0" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="implicit"/>
</element>
<element type="quadricpart" label="g">
<show object="true" label="false" ev="4"/>
<condition showObject="c"/>
<objColor r="0" g="0" b="255" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="0.0" x1="0.0" y1="1.0" z1="0.0" x2="0.0" y2="0.0" z2="1.0"/>
</element>
<element type="conic3d" label="d">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-36.0" A3="0.0" A4="-0.0" A5="-0.0"/>
<eqnStyle style="implicit"/>
</element>
<command name="Sphere">
<input a0="(0, 0)" a1="A"/>
<output a0="h_1"/>
</command>
<element type="quadric" label="h_1">
<show object="true" label="false" ev="4"/>
<condition showObject="c"/>
<objColor r="0" g="100" b="0" alpha="0.3799999952316284"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="0.0" x1="0.0" y1="1.0" z1="0.0" x2="0.0" y2="0.0" z2="1.0"/>
<matrix A0="1.0" A1="1.0" A2="1.0" A3="-36.0" A4="0.0" A5="0.0" A6="0.0" A7="-0.0" A8="-0.0" A9="-0.0"/>
</element>
<element type="boolean" label="h">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="17" y="30"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="StartAnimation(k,h) If(h,SetValue(c,false))"/>
<caption val="球體表面積"/>
</element>
<element type="boolean" label="c">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="13" y="395"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(c,SetValue(h,false))"/>
<caption val="球冠表面積"/>
</element>
</construction>
</geogebra>