Area of the Curved Surface of a Cone
2014-09-28 04:05
This GeoGebra applet dynamically demonstrates the geometric unfolding of a cone's lateral surface into a circular sector, illustrating the derivation of the curved surface area formula. Using commands such as Sequence, CircleArc, and CircleSector, it constructs a 3D cone model alongside its unfolded net, revealing the correspondence between the cone's base radius and slant height with the sector's
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.2" version="4.2.56.0" id="6a15b6da-723c-4111-8742-4253b4b90992" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="963" height="635" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.0" orientation="0" />
</panes>
<views>
<view id="4" toolbar="0 59 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="2,2" size="200" window="0,0,400,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="2" size="479" window="0,0,400,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="0" size="200" window="0,0,200,400" />
<view id="8" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<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" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="4097" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 59 | 1 501 67 , 5 19 , 72 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="18"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="955" height="452"/>
<coordSystem xZero="134.15919973026104" yZero="361.943511074585" scale="118.83798673353132" yscale="118.83798673353132"/>
<evSettings axes="false" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="13" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="255" g="255" b="255"/>
<lineStyle axes="1" grid="10"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<grid distX="0.5" distY="0.5" distTheta="0.5235987755982988"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<command name="Point">
<input a0="CircleArc[(0, 1), (1, 1), (0, 0)]"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false"/>
<condition showObject="k ≟ 0 ∨ (¬分割扇形)"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-27" y="15"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-0.2971633146824652" y="1.9548266672055878" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="If">
<input a0="y(A) > 0" a1="Angle[(0, 0), (0, 1), A]" a2="6.283185307179586"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<value val="3.443313038165757"/>
<show object="false" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<allowReflexAngle val="true"/>
</element>
<element type="numeric" label="n">
<value val="6.0"/>
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="3" max="100" absoluteScreenLocation="true" width="100.0" x="372.0" y="460.0" fixed="true" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="θ" exp="α / (2 * n)" />
<element type="angle" label="θ">
<value val="0.28694275318047974"/>
<allowReflexAngle val="true"/>
</element>
<command name="Sequence">
<input a0="(sin(i * θ), 1 - cos(i * θ))" a1="i" a2="0" a3="2 * n"/>
<output a0="list1"/>
</command>
<element type="list" label="list1">
<show object="false" label="false"/>
<objColor r="0" g="110" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="l" exp="2 * sin(θ / 2)" />
<element type="numeric" label="l">
<value val="0.2859593591816908"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<element type="point" label="O">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="E">
<show object="false" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="2.0" y="-0.5" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="E" a1="(x(E) + 1, y(E))"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="1.0" z="0.5"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="a"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="2.0" y="-0.5" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Min">
<input a0="2 * (x(F) - x(E))" a1="1"/>
<output a0="k"/>
</command>
<element type="numeric" label="k">
<value val="0.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Max">
<input a0="2 * (x(F) - x(E)) - 1" a1="0"/>
<output a0="k'"/>
</command>
<element type="numeric" label="k'">
<value val="0.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="φ" exp="(1 - k) * θ / 2" />
<element type="angle" label="φ">
<value val="0.14347137659023987"/>
<allowReflexAngle val="true"/>
</element>
<command name="Sequence">
<input a0="If[k < 1, (x(O) + l * sin(i * φ) * cos(i * φ) / sin(φ), y(O) + l * sin(i * φ) * sin(i * φ) / sin(φ)), (x(O) + l * i, y(O))]" a1="i" a2="0" a3="2 * n"/>
<output a0="list2"/>
</command>
<element type="list" label="list2">
<show object="false" label="false"/>
<objColor r="0" g="110" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Sector[Rotate[Translate[Circle[(x(O) + 0, y(O) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], -(2 * i - 1) * k * θ / 2, Element[list2, i]], Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], -(2 * i - 1) * k * θ / 2, Element[list2, i]], Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], -(2 * i - 1) * k * θ / 2, Element[list2, i]]]" a1="i" a2="1" a3="n"/>
<output a0="list3"/>
</command>
<element type="list" label="list3">
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Sector[Rotate[Rotate[Translate[Circle[(x(O) + 0, y(O) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], -(2 * i - 1) * k * θ / 2, Element[list2, i]], 180° * k', (x(O) + n * l - sin(θ / 2) / 2, y(O) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], -(2 * i - 1) * k * θ / 2, Element[list2, i]], 180° * k', (x(O) + n * l - sin(θ / 2) / 2, y(O) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], -(2 * i - 1) * k * θ / 2, Element[list2, i]], 180° * k', (x(O) + n * l - sin(θ / 2) / 2, y(O) + cos(θ / 2) / 2)]]" a1="i" a2="n + 1" a3="2 * n"/>
<output a0="list4"/>
</command>
<element type="list" label="list4">
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="m">
<value val="0.0"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="100.0" x="305.0" y="69.0" fixed="true" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<command name="If">
<input a0="y(A) > 0" a1="CircleArc[(0, 1), (0, 0), A]" a2="6.283185307179586"/>
<output a0="s"/>
</command>
<element type="conicpart" label="s">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" 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="0.0" A3="0.0" A4="-0.0" A5="-1.0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="L" exp="(sin((1 - m) * s) / (1 - m), (1 - cos((1 - m) * s)) / (1 - m))" type="point" />
<element type="point" label="L">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.2971633146824654" y="1.9548266672055876" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="CircleArc">
<input a0="(0, 1 / (1 - m))" a1="O" a2="L"/>
<output a0="b"/>
</command>
<element type="conicpart" label="b">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="3"/>
<lineStyle thickness="6" 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="0.0" A3="0.0" A4="-0.0" A5="-1.0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="B" exp="(x(O) + l * sin(n * (1 - m) * θ / 2) * cos(n * (1 - m) * θ / 2) / sin((1 - m) * θ / 2), y(O) + l * sin(n * (1 - m) * θ / 2) * sin(n * (1 - m) * θ / 2) / sin((1 - m) * θ / 2))" type="point" />
<element type="point" label="B">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.9886421666117595" y="1.1502886103375967" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="text1" exp=""2π \ell""/>
<element type="text" label="text1">
<show object="true" label="false"/>
<condition showObject="y(A) ≟ 0"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.0" size="0" style="0"/>
<startPoint exp="B"/>
<labelOffset x="7" y="-31"/>
</element>
<expression label="text2" exp=""2πr""/>
<element type="text" label="text2">
<show object="true" label="false"/>
<condition showObject="y(A) > 0"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.0" size="0" style="0"/>
<startPoint exp="B"/>
<labelOffset x="8" y="-30"/>
</element>
<expression label="text3" exp=""2πr""/>
<element type="text" label="text3">
<show object="true" label="false"/>
<condition showObject="m ≟ 1 ∧ (y(A) > 0)"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="B'"/>
<labelOffset x="1" y="20"/>
</element>
<expression label="text4" exp=""2π\ell""/>
<element type="text" label="text4">
<show object="true" label="false"/>
<condition showObject="m ≟ 1 ∧ y(A) ≟ 0"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.0" size="0" style="0"/>
<startPoint exp="B'"/>
</element>
<command name="Segment">
<input a0="(0, 0)" a1="(s, 0)"/>
<output a0="d"/>
</command>
<element type="segment" label="d">
<show object="true" label="false"/>
<condition showObject="m ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="3.4433130381657575" z="0.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
<input a0="d"/>
<output a0="B'"/>
</command>
<element type="point" label="B'">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.7216565190828788" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="CircleArc">
<input a0="(0, 1)" a1="A" a2="(0, 0)"/>
<output a0="c"/>
</command>
<element type="conicpart" label="c">
<show object="true" label="false"/>
<condition showObject="k ≟ 0 ∨ (¬分割扇形)"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="10" 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="0.0" A3="0.0" A4="-0.0" A5="-1.0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
<input a0="a"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="204" g="204" b="204" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.5" y="-0.5" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<command name="Point">
<input a0="Segment[(0, 1), (s, 1)]"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="1.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<command name="Segment">
<input a0="D" a1="(x(D), y(D) - 1)"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.0" y="0.0" z="0.0"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
<input a0="e"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="0.5" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="text5" exp=""\ell""/>
<element type="text" label="text5">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.0" size="0" style="0"/>
<startPoint exp="G"/>
<labelOffset x="-17" y="-22"/>
</element>
<expression label="text6" exp=""圓錐的曲面面積""/>
<element type="text" label="text6">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.3333333333333333" size="6" style="1"/>
<absoluteScreenLocation x="17" y="36"/>
</element>
<expression label="text7" exp=""將弧拉直""/>
<element type="text" label="text7">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.1111111111111112" size="2" style="0"/>
<absoluteScreenLocation x="218" y="75"/>
</element>
<command name="CircleSector">
<input a0="(0, 1)" a1="(0, 0)" a2="A"/>
<output a0="f"/>
</command>
<element type="conicpart" label="f">
<show object="true" label="false"/>
<condition showObject="(¬分割扇形) ∧ (y(A) > 0)"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="1" 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="0.0" A3="0.0" A4="-0.0" A5="-1.0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text8" exp=""重新排列""/>
<element type="text" label="text8">
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.1111111111111112" size="2" style="0"/>
<absoluteScreenLocation x="272" y="426"/>
</element>
<command name="Circle">
<input a0="(0, 1)" a1="1"/>
<output a0="g"/>
</command>
<element type="conic" label="g">
<show object="true" label="false"/>
<condition showObject="y(A) ≟ 0 ∧ (¬分割扇形)"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" 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="0.0" A3="0.0" A4="-0.0" A5="-1.0"/>
<eqnStyle style="specific"/>
</element>
<element type="boolean" label="分割扇形">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="37" y="417"/>
<labelMode val="0"/>
</element>
<command name="Midpoint">
<input a0="A" a1="D"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.1485816573412326" y="1.477413333602794" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="text10" exp=""\ell""/>
<element type="text" label="text10">
<show object="true" label="false"/>
<condition showObject="(y(A) > 0) ∧ k ≟ 0"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.0" size="0" style="0"/>
<startPoint exp="H"/>
<labelOffset x="-14" y="1"/>
</element>
<expression label="text11" exp=""分割圓錐曲面的扇形，將弧拉直。""/>
<element type="text" label="text11">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="498" y="72"/>
</element>
<expression label="text12" exp=""重新排列分割部份，增加分割份數。""/>
<element type="text" label="text12">
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="498" y="101"/>
</element>
<expression label="text13" exp=""隨著分割份數增加，圖形愈來愈接近一個什麼形狀？""/>
<element type="text" label="text13">
<show object="true" label="false"/>
<condition showObject="h ∧ 分割扇形"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="498" y="130"/>
</element>
<expression label="text14" exp=""圓錐曲面的面積是多少？""/>
<element type="text" label="text14">
<show object="true" label="false"/>
<condition showObject="h ∧ i ∧ 分割扇形"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="498" y="159"/>
</element>
<element type="boolean" label="h">
<value val="false"/>
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="465" y="113"/>
<labelMode val="0"/>
</element>
<element type="boolean" label="i">
<value val="false"/>
<show object="true" label="false"/>
<condition showObject="h ∧ 分割扇形"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="465" y="142"/>
<labelMode val="0"/>
</element>
<expression label="text9" exp=""將扇形分成 " + (2 * n) + " 份"" />
<element type="text" label="text9">
<show object="true" label="false"/>
<condition showObject="分割扇形"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.1111111111111112" size="2" style="0"/>
<absoluteScreenLocation x="199" y="466"/>
</element>
<element type="numeric" label="j">
<value val="0.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="100.0" x="819.0" y="34.0" fixed="false" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<expression label="I" exp="(x(Corner[3] - j), y(Corner[3] - j))" type="point" />
<element type="point" label="I">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.915640552818693" y="3.0541034988114357" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="text15" exp=""重置""/>
<element type="text" label="text15">
<show object="true" label="false"/>
<objColor r="0" g="102" b="153" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<startPoint exp="I"/>
<labelOffset x="-62" y="17"/>
</element>
</construction>
</geogebra>