Curved Surface Area of Cones (Parametric Curved Surface)
2014-10-21 09:07
This GeoGebra applet demonstrates how the curved surface area of a cone unfolds into a circular sector. Using parametric surfaces and geometric transformations, it visually illustrates the unfolding process and the intrinsic connection to the sector area formula, helping students deeply understand the calculation principles of the curved surface area of a cone in solid geometry.
In collections
S3 Area and Volume
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.280.0" id="6a05659c-e9e4-4d4f-ba82-77482b8d7eee" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="898" height="548" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.3518930957683742" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="162,92,700,550" />
<view id="1" visible="false" inframe="false" stylebar="false" location="1,3" size="288" window="100,100,600,400" />
<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="2" visible="false" inframe="false" stylebar="false" location="3" size="316" 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="128" visible="false" inframe="true" stylebar="false" location="1" size="480" 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="43" visible="false" inframe="false" stylebar="false" location="1" size="450" 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 , 22 , 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="true" location="1" size="898" window="100,100,600,400" />
</views>
<toolbar show="false" items="0 39 | 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 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="576" height="401"/>
<coordSystem xZero="215" yZero="314.99999999999983" scale="50" yscale="49.99999999999998"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="10"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" 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"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.9802847263105987" yZero="-1.2122206907701296" zZero="-0.9208628020238414" scale="60.6827322060104" xAngle="26" zAngle="-47"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="1" rightAngleStyle="1" gridType="0"/>
<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="true" show="false" size="2"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<command name="Point">
<input a0="zAxis"/>
<output a0="A"/>
</command>
<element type="point3d" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0" y="0" z="3" w="1"/>
<pointSize val="3"/>
</element>
<command name="Point">
<input a0="Segment[A, (0, 0, 0)]"/>
<output a0="P"/>
</command>
<element type="point3d" label="P">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="10" y="16"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0" y="0" z="3" w="1"/>
<pointSize val="5"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="2.337840536707809" y="0" z="1"/>
<coordStyle style="cartesian3d"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="P" a1="Distance[A, B]" a2="yAxis"/>
<output a0="c"/>
</command>
<element type="conic3d" label="c">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-14.465498375074255" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="xOyPlane" a1="c"/>
<output a0="C" a1="D"/>
</command>
<element type="point3d" label="C">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.337840536707809" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<element type="point3d" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.3378405367078083" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<expression label="α" exp="((2 * π) * x(B) / x(D))" />
<element type="numeric" label="α">
<value val="6.283185307179588"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="γ" exp="π - α / 2" />
<element type="numeric" label="γ">
<value val="-8.881784197001252e-16"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="1"/>
<coordStyle style="cartesian3d"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="f" exp="f(x) = (((-z(P))) / x(D) * (x - x(D)))" />
<element type="function" label="f">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Rotate">
<input a0="D" a1="γ" a2="zAxis"/>
<output a0="D_1"/>
</command>
<element type="point3d" label="D_1">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.3378405367078083" y="-2.076419513404034e-15" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Rotate">
<input a0="D_1" a1="α" a2="zAxis"/>
<output a0="D'"/>
</command>
<element type="point3d" label="D'">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.3378405367078083" y="1.5038137273648038e-15" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Rotate">
<input a0="D_1" a1="(π * x(B) / x(D))" a2="zAxis"/>
<output a0="D'_1"/>
</command>
<element type="point3d" label="D'_1">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.3378405367078083" y="2.86302893019615e-16" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Mirror">
<input a0="D'_1" a1="O"/>
<output a0="E"/>
</command>
<element type="point3d" label="E">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.3378405367078083" y="-2.86302893019615e-16" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="P" a1="D'"/>
<output a0="e"/>
</command>
<element type="segment3d" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Circle">
<input a0="zAxis" a1="B"/>
<output a0="g"/>
</command>
<element type="conic3d" label="g">
<show object="true" label="false" ev="4"/>
<condition showObject="P ≟ A"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="2"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-5.465498375074255" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
</element>
<command name="CircumcircleArc">
<input a0="D_1" a1="E" a2="D'"/>
<output a0="h"/>
</command>
<element type="conicpart" label="h">
<show object="true" label="false" ev="4"/>
<condition showObject="P ≟ O"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="15" typeHidden="2"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="1" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CircumcircleArc">
<input a0="D_1" a1="D'_1" a2="D'"/>
<output a0="d"/>
</command>
<element type="conicpart" label="d">
<show object="true" label="false" ev="4"/>
<condition showObject="P ≠ A"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="2"/>
<eigenvectors x0="-1" y0="5.053215498074303e-16" z0="1.0" x1="-5.053215498074303e-16" y1="-1" z1="1.0"/>
<matrix A0="0" A1="0" A2="0" A3="0" A4="-2.5266077490371514e-16" A5="-0.5"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="P" a1="D_1"/>
<output a0="b"/>
</command>
<element type="segment3d" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Surface">
<input a0="(u * cos(v))" a1="(u * sin(v))" a2="f(u)" a3="u" a4="0" a5="x(D)" a6="v" a7="γ" a8="(2 * π) - γ"/>
<output a0="a"/>
</command>
<element type="surfacecartesian3d" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.65"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="Vector">
<input a0="A + (0, 0, 1)" a1="A + (0, 0, 0.2)"/>
<output a0="u"/>
</command>
<element type="vector3d" label="u">
<show object="true" label="true" ev="4"/>
<condition showObject="P ≠ O"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-77" y="-23"/>
<labelMode val="3"/>
<coords x="0" y="0" z="-0.7999999999999998" w="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint x="0" y="0" z="4" w="1"/>
<caption val="Press the red point"/>
</element>
<expression label="text1" exp=""2 π r""/>
<element type="text" label="text1">
<show object="true" label="false" ev="4"/>
<condition showObject="P ≠ A"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="D'_1"/>
<labelOffset x="-53" y="-20"/>
</element>
<expression label="text2" exp=""l""/>
<element type="text" label="text2">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="Midpoint[b]"/>
</element>
<expression label="text3" exp=""l""/>
<element type="text" label="text3">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="Midpoint[e]"/>
</element>
<command name="Segment">
<input a0="O" a1="A"/>
<output a0="i"/>
</command>
<element type="segment3d" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="O" a1="B"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false" ev="4"/>
<condition showObject="P ≟ A"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="2.337840536707809" z="0"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text4" exp=""h""/>
<element type="text" label="text4">
<show object="true" label="false" ev="4"/>
<condition showObject="P ≟ A"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="Midpoint[i]"/>
<labelOffset x="6" y="46"/>
</element>
<expression label="text5" exp=""r""/>
<element type="text" label="text5">
<show object="true" label="false" ev="4"/>
<condition showObject="P ≟ A"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="Midpoint[j]"/>
<labelOffset x="10" y="-1"/>
</element>
</construction>
</geogebra>