Exploring the Area of a Circle Formula
2011-09-07 08:41
This example demonstrates the derivation of the circle area formula by cutting a circle into sectors and rearranging them into a parallelogram-like figure. As the number of sectors increases, the shape approaches a parallelogram, helping students visually understand the relationship between the area, radius, and circumference.
In collections
S3 Area and Volume
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.139.0" id="cf419658-3f48-4eda-9841-bf81ad5f255b" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="989" height="724" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.20222446916076844" 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="false" location="1,1" size="200" window="0,0,400,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="971" window="0,0,400,400" />
<view id="2" visible="false" inframe="true" stylebar="true" location="3" size="200" window="26,208,314,529" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 6" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="16" visible="false" inframe="true" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="4097" visible="false" inframe="true" stylebar="true" location="1" size="150" window="492,251,697,500" />
<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" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 59 || 1 501 5 19 67 72 | 2 15 45 18 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 70 65 | 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 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6" position="1" help="true" />
<input show="true" cmd="true" top="true" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="18"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="971" height="525"/>
<coordSystem xZero="117.25120895357163" yZero="422.3303776334349" scale="89.13530378395939" yscale="89.13530378395951"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="1" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="224" g="224" b="224"/>
<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="2.0" 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="教育局 評估及考評局組" date="">
<element type="numeric" label="n">
<value val="6.0"/>
<show object="true" label="false"/>
<condition showObject="dissect"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="3" max="100" absoluteScreenLocation="true" width="200.0" x="481.0" y="505.0" fixed="false" horizontal="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="θ" exp="(360 * ° / ((2 * n)))" />
<element type="angle" label="θ">
<angleStyle val="0"/>
<value val="0.5235987755982988"/>
<show object="false" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<expression label="l" exp="(2 * sin(θ / 2))" />
<element type="numeric" label="l">
<value val="0.5176380902050415"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<element type="point" label="E">
<show object="false" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="1"/>
<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="2"/>
<pointStyle val="1"/>
</element>
<command name="Segment">
<input a0="E" a1="(x(E) + 4, y(E))"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="true" label="false" ev="4"/>
<condition showObject="dissect"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0.0" y="4.0" z="2.0"/>
<lineStyle thickness="4" 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" ev="4"/>
<condition showObject="dissect"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="1"/>
<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="4"/>
<pointStyle val="0"/>
</element>
<command name="Min">
<input a0="(x(F) - x(E)) / (a / 2)" 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>
<expression label="φ" exp="((1 - k) * θ / 2)" />
<element type="angle" label="φ">
<angleStyle val="0"/>
<value val="0.2617993877991494"/>
<show object="false" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="1"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Point">
<input a0="yAxis"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="false" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="1"/>
<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="4"/>
<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="150.0" x="198.0" y="95.0" fixed="true" horizontal="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</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="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="(x(O) + sin((i * θ)), y(O) + 1 - cos((i * θ)))" a1="i" a2="0" a3="(2 * n)"/>
<output a0="list1"/>
</command>
<element type="list" label="list1">
<show object="false" label="true"/>
<objColor r="0" g="110" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Max">
<input a0="(x(F) - x(E)) / (a / 2) - 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>
<command name="Rotate">
<input a0="(x(O) + 0, y(O) + 1)" a1="((-k) * θ / 2)" a2="O"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="0.0" y="1.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="O" a1="((-k) * θ / 2)" a2="O"/>
<output a0="O'"/>
</command>
<element type="point" label="O'">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="Element[list1, 1]" a1="((-k) * θ / 2)" a2="O"/>
<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="1"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Sector">
<input a0="Rotate[Circle[(x(O) + 0, y(O) + 1), O], ((-k) * θ / 2), O]" a1="Rotate[O, ((-k) * θ / 2), O]" a2="Rotate[Element[list1, 1], ((-k) * θ / 2), O]"/>
<output a0="c"/>
</command>
<element type="conicpart" label="c">
<show object="false" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="3" 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"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Rotate">
<input a0="(x(O) + 0, y(O) + 1)" a1="((-k) * θ / 2)" a2="O"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="0.0" y="1.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="O" a1="((-k) * θ / 2)" a2="O"/>
<output a0="O'_1"/>
</command>
<element type="point" label="O'_1">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="Element[list1, 1]" a1="((-k) * θ / 2)" a2="O"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Arc">
<input a0="Rotate[Circle[(x(O) + 0, y(O) + 1), O], ((-k) * θ / 2), O]" a1="O'_1" a2="D"/>
<output a0="d"/>
</command>
<element type="conicpart" label="d">
<show object="false" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="7" 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"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CurveCartesian">
<input a0="t - sin(t)" a1="1 - cos(t)" a2="t" a3="0" a4="6.28319"/>
<output a0="b"/>
</command>
<element type="curvecartesian" label="b">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<element type="point" label="G">
<show object="true" label="false"/>
<condition showObject="dissect ≟ false"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0.0" y="1.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<command name="Circle">
<input a0="(0, 1)" a1="1"/>
<output a0="h"/>
</command>
<element type="conic" label="h">
<show object="true" label="false"/>
<condition showObject="dissect ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<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="specific"/>
</element>
<element type="boolean" label="dissect">
<value val="false"/>
<show object="true" label="true"/>
<condition showObject="m ≟ 1 ∧ circumference"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelOffset x="39" y="447"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="circumference">
<value val="false"/>
<show object="true" label="true"/>
<condition showObject="m ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelOffset x="688" y="388"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="Ray">
<input a0="(0, 0)" a1="(1, 0)"/>
<output a0="i"/>
</command>
<element type="ray" label="i">
<show object="true" label="false"/>
<objColor r="153" g="153" b="153" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0.0" y="1.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text3" exp=""The area of a circle""/>
<element type="text" label="text3">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.2222222222222223" size="4" style="1"/>
<auxiliary val="false"/>
<absoluteScreenLocation x="21" y="36"/>
</element>
<command name="Point">
<input a0="Segment[(0, 1), (6.28319, 1)]"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="false"/>
<condition showObject="dissect"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="2"/>
<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="5"/>
<pointStyle val="1"/>
</element>
<expression label="text7" exp=""Rearrange the dissected parts. Increase its number. What shape does the figure tend to?""/>
<element type="text" label="text7">
<show object="true" label="false"/>
<condition showObject="circumference ∧ dissect"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.1111111111111112" size="2" style="0"/>
<auxiliary val="false"/>
<absoluteScreenLocation x="413" y="35"/>
</element>
<command name="Segment">
<input a0="H" a1="(x(H), 0)"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false"/>
<condition showObject="dissect"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="1.0" y="0.0" z="0.0"/>
<lineStyle thickness="3" type="10" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
<input a0="j"/>
<output a0="I"/>
</command>
<element type="point" label="I">
<show object="false" label="false"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="0.0" y="0.5" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="text6" exp=""r""/>
<element type="text" label="text6">
<show object="true" label="false"/>
<condition showObject="dissect"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<startPoint exp="I"/>
<labelOffset x="-17" y="-1"/>
</element>
<expression label="text9" exp=""rearrange""/>
<element type="text" label="text9">
<show object="true" label="false"/>
<condition showObject="dissect"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.1111111111111112" size="2" style="0"/>
<auxiliary val="false"/>
<startPoint x="0.737629067891954" y="-0.5348006944824433" z="1.0"/>
</element>
<command name="Midpoint">
<input a0="a"/>
<output a0="J"/>
</command>
<element type="point" label="J">
<show object="true" label="false"/>
<condition showObject="dissect"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="4.0" y="-0.5" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<element type="point" label="K">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="6.283185307179586" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="text1" exp=""circumference = ""/>
<element type="text" label="text1">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="false"/>
<absoluteScreenLocation x="682" y="411"/>
</element>
<expression label="text2" exp=""= 2 π r""/>
<element type="text" label="text2">
<show object="true" label="false"/>
<condition showObject="circumference ∧ m ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="false"/>
<absoluteScreenLocation x="844" y="410"/>
</element>
<expression label="L" exp="(sin(((2 * 3.141592653589793) * (1 - m))) / (1 - m), (1 - cos(((2 * 3.141592653589793) * (1 - m)))) / (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="2"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="L" a1="1 / (1 - m)"/>
<output a0="p"/>
</command>
<element type="conic" label="p">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<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="specific"/>
</element>
<command name="Circle">
<input a0="(0, 0)" a1="1 / (1 - m)"/>
<output a0="q"/>
</command>
<element type="conic" label="q">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<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="specific"/>
</element>
<command name="Intersect">
<input a0="p" a1="q" a2="1"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="false" label="true"/>
<objColor r="0" g="102" b="102" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="(0, 1)" a1="1"/>
<output a0="c_1"/>
</command>
<element type="conic" label="c_1">
<show object="true" label="false"/>
<condition showObject="m ≟ 0"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="7" 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>
<command name="Segment">
<input a0="(0, 0)" a1="(6.283185307179586, 0)"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="false"/>
<condition showObject="m ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0.0" y="6.283185307179586" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="G" a1="(x(G), y(G) - 1)"/>
<output a0="r"/>
</command>
<element type="segment" label="r">
<show object="true" label="true"/>
<condition showObject="dissect ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelOffset x="-1" y="0"/>
<labelMode val="0"/>
<coords x="1.0" y="0.0" z="0.0"/>
<lineStyle thickness="3" type="10" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text8" exp=""straighten the circumference""/>
<element type="text" label="text8">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="false"/>
<absoluteScreenLocation x="67" y="88"/>
</element>
<element type="boolean" label="o1">
<value val="false"/>
<show object="true" label="false"/>
<condition showObject="circumference ∧ dissect"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelOffset x="370" y="72"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="o2">
<value val="false"/>
<show object="true" label="false"/>
<condition showObject="circumference ∧ dissect ∧ o1"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelOffset x="370" y="105"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="text11" exp=""What are the dimensions of this shape?""/>
<element type="text" label="text11">
<show object="true" label="false"/>
<condition showObject="circumference ∧ dissect ∧ o1"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.1111111111111112" size="2" style="0"/>
<auxiliary val="false"/>
<absoluteScreenLocation x="413" y="96"/>
</element>
<expression label="text12" exp=""What is the area of the circle?""/>
<element type="text" label="text12">
<show object="true" label="false"/>
<condition showObject="circumference ∧ dissect ∧ o1 ∧ o2"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.1111111111111112" size="2" style="0"/>
<auxiliary val="false"/>
<absoluteScreenLocation x="413" y="129"/>
</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="true"/>
<condition showObject="dissect ≟ true"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="2"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="4"/>
<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="true"/>
<condition showObject="dissect ≟ true"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="2"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="CircleArc">
<input a0="(0, 1 / (1 - m))" a1="O" a2="L"/>
<output a0="s"/>
</command>
<element type="conicpart" label="s">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<lineStyle thickness="7" 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>
<element type="numeric" label="f">
<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="673.0" y="57.0" fixed="false" horizontal="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<expression label="P" exp="(x(Corner[3] - f), y(Corner[3] - f))" type="point" />
<element type="point" label="P">
<show object="false" label="false"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="9.589340640136431" y="4.749300890469574" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="text4" exp=""\text{dissect the circle into } \fgcolor{red}{" + (LaTeX[(2 * n)]) + " } \text{ parts}"" />
<element type="text" label="text4">
<show object="true" label="false"/>
<condition showObject="dissect"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.1111111111111112" size="2" style="0"/>
<auxiliary val="false"/>
<absoluteScreenLocation x="165" y="489"/>
</element>
</construction>
</geogebra>