Surface of Revolution: Generating a Cone from a Rotating Triangle
2020-07-19 05:55
This GeoGebra applet demonstrates generating a solid of revolution. It visualizes how a right triangle forms a cone when rotated around a leg. Using sliders for rotation angle and radius, with polygon and surface commands, it presents the generation mechanism and spatial structure, aiding students in developing spatial imagination in solid geometry.
In collections
Minority Pi GeoGebra Collection
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.593.0" app="classic" platform="d" id="7b088cf1-a71e-40eb-9fe6-8dbc89955c1e" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1366" height="728" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.1800878477306003" orientation="1" />
<pane location="1" divider="0.328696925329429" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="3,1,1" size="372" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1,3" size="449" 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="16" visible="false" inframe="false" stylebar="true" location="3,1" size="544" 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 51 | 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 , 538 | 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="1,1" size="634" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="246" window="100,100,600,400" />
<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="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="239" window="100,100,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</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="true" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="449" height="497"/>
<coordSystem xZero="132.53515746926175" yZero="347.38035079675655" scale="66.87350179509912" yscale="66.87350179509896"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="0" g="51" b="0"/>
<axesColor r="255" g="255" b="255"/>
<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>
<algebraView>
<mode val="1"/>
</algebraView>
<kernel>
<uses3D val="true"/>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.778013859063017" yZero="-0.5458393541789994" zZero="-1.2386854072851121" scale="61.89014038897301" xAngle="24.0" zAngle="-43.0"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<plate show="true"/>
<bgColor r="0" g="51" b="0"/>
<clipping use="false" show="false" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="numeric" label="r">
<value val="1.29"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="120.0" x="67.0" y="42.0" fixed="false" horizontal="true" showAlgebra="true"/>
<pointSize val="3.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<ggbscript onUpdate=""/>
</element>
<element type="numeric" label="h">
<value val="2.83"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="120.0" x="65.0" y="77.0" fixed="false" horizontal="true" showAlgebra="true"/>
<pointSize val="3.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<ggbscript onUpdate=""/>
</element>
<element type="angle" label="α">
<angleStyle val="3"/>
<value val="6.283185307179587"/>
<slider min="0" max="6.283185307179586" absoluteScreenLocation="true" width="120.0" x="66.0" y="119.0" fixed="false" horizontal="true" showAlgebra="true"/>
<pointSize val="3.0"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="6" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation step="0.01°" speed="1" type="0" playing="true"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-10" y="1"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="0.0" y="-0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="A" exp="(r, 0, 0)" type="point" />
<element type="point3d" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="6" y="9"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="1.29" y="0.0" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<expression label="B" exp="(0, 0, h)" type="point" />
<element type="point3d" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="0.0" y="0.0" z="2.83" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Polygon">
<input a0="B" a1="O" a2="A"/>
<output a0="t1" a1="a" a2="b" a3="o"/>
</command>
<element type="polygon3d" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.15000000596046448"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<element type="segment3d" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="o">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Rotate">
<input a0="Polygon[B, O, A]" a1="α" a2="O"/>
<output a0="t2"/>
</command>
<element type="polygon3d" label="t2">
<lineStyle thickness="3" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<command name="CircleArc">
<input a0="O" a1="A" a2="Rotate[A, α, O]" a3="xOyPlane"/>
<output a0="d"/>
</command>
<element type="conicpart" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<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="1.0" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="parametric"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Circle">
<input a0="O" a1="r"/>
<output a0="e"/>
</command>
<element type="conic" label="e">
<show object="true" label="false" ev="4"/>
<condition showObject="α ≟ 360°"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<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="-1.6641000000000001" A3="0.0" A4="-0.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Surface">
<input a0="((r - (u * r / h)) * cos(θ))" a1="((r - (u * r / h)) * sin(θ))" a2="u" a3="θ" a4="0" a5="α" a6="u" a7="0" a8="h"/>
<output a0="c"/>
</command>
<element type="surfacecartesian3d" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
</construction>
</geogebra>