Circle Circumference Formula Application 3: Arc and Semicircle Integrals
2014-01-30 12:10
This GeoGebra applet demonstrates the circle circumference formula's application, focusing on constructing arcs, semicircles, and segments. Using Integral commands, it illustrates calculus methods for calculating circle-related areas or lengths. Ideal for teaching geometry and calculus, it helps students intuitively grasp circle formulas and integrals in geometric contexts.
In collections
Circle Circumference
Loading the math board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.284.0" id="c68269c4-c6fb-40c6-832a-a97e0e7075cd" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="893" height="607" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.22396416573348266" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="218,323,972,581" />
<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="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 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="1" size="891" window="100,100,600,400" />
<view id="2" visible="false" inframe="true" stylebar="true" location="3" size="200" window="940,91,325,739" />
<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 , 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="false" inframe="true" stylebar="false" location="1" size="480" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 | 40 | 62 | 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="890" height="550"/>
<coordSystem xZero="209.8980987711812" yZero="216.80978791900134" scale="80.6442020306206" yscale="80.6442020306206"/>
<evSettings axes="false" grid="false" 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="192" g="192" b="192"/>
<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="1" distY="1" distTheta="0.5235987755982988"/>
</euclidianView>
<kernel>
<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"/>
<construction title="" author="" date="30 January 2014">
<command name="CircleArc">
<input a0="(0, 0)" a1="(2, 0)" a2="(0, 2)"/>
<output a0="c"/>
</command>
<element type="conicpart" label="c">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<ggbscript val="SetValue[ShowArc, !ShowArc] SetValue[ShowArcMajor, true] SetLineThickness[c, If[ShowArc, 8, 4]] SetColor[c, If[ShowArc, "Red", "Black"]]"/>
<lineStyle thickness="4" 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="-4" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Semicircle">
<input a0="(0, 2)" a1="(0, 0)"/>
<output a0="d"/>
</command>
<element type="conicpart" label="d">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<ggbscript val="SetValue[ShowSemicircle, !ShowSemicircle] SetValue[ShowSemicircle2, true] SetLineThickness[d, If[ShowSemicircle, 8, 4]] SetColor[d, If[ShowSemicircle, "Red", "Black"]]"/>
<lineStyle thickness="4" 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="0" A3="0" A4="0" A5="-1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="(0, 0)" a1="(2, 0)"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="-23" y="18"/>
<labelMode val="3"/>
<fixed val="true"/>
<ggbscript val="SetValue[ShowSegment, !ShowSegment] SetLineThickness[a, If[ShowSegment, 8, 4]] SetColor[a, If[ShowSegment, "Red", "Black"]]"/>
<coords x="0" y="2" z="0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="20 cm"/>
</element>
<element type="numeric" label="c_1">
<value val="0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
</element>
<expression label ="f" exp="f(x) = sqrt(4 - x^(2))"/>
<element type="function" label="f">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<expression label ="g" exp="g(x) = 1 + sqrt(1 - x^(2))"/>
<element type="function" label="g">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<expression label ="h" exp="h(x) = 1 - sqrt(1 - x^(2))"/>
<element type="function" label="h">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Integral">
<input a0="f" a1="1" a2="2"/>
<output a0="b"/>
</command>
<element type="numeric" label="b">
<value val="1.228369698608759"/>
<show object="true" label="false"/>
<objColor r="255" g="215" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectionAllowed val="false"/>
</element>
<element type="numeric" label="c_2">
<value val="0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
</element>
<command name="IntegralBetween">
<input a0="f" a1="g" a2="0" a3="1"/>
<output a0="e"/>
</command>
<element type="numeric" label="e">
<value val="0.12782478915118212"/>
<show object="true" label="false"/>
<objColor r="255" g="215" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectionAllowed val="false"/>
</element>
<element type="numeric" label="c_3">
<value val="0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
</element>
<element type="numeric" label="c_4">
<value val="0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
</element>
<command name="Integral">
<input a0="h" a1="0" a2="1"/>
<output a0="i"/>
</command>
<element type="numeric" label="i">
<value val="0.21460183475215644"/>
<show object="true" label="false"/>
<objColor r="255" g="215" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectionAllowed val="false"/>
</element>
<element type="numeric" label="c_5">
<value val="0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
</element>
<element type="boolean" label="ShowArc">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="ShowSemicircle">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="ShowSegment">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="CircleArc">
<input a0="(0, 0)" a1="(0, 2)" a2="(2, 0)"/>
<output a0="k"/>
</command>
<element type="conicpart" label="k">
<show object="true" label="false"/>
<condition showObject="ShowArcMajor ∧ ShowArc"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<ggbscript val="SetValue[ShowArcMajor, !ShowArcMajor] "/>
<lineStyle thickness="3" type="10" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-4" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""\frac{1}{4}""/>
<element type="text" label="text1">
<show object="true" label="false"/>
<condition showObject="ShowArc"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="516" y="52"/>
</element>
<expression label="text2" exp=""個圓周；""/>
<element type="text" label="text2">
<show object="true" label="false"/>
<condition showObject="ShowArc"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="544" y="92"/>
</element>
<expression label="text1_1" exp=""\frac{1}{4}""/>
<element type="text" label="text1_1">
<show object="true" label="false"/>
<condition showObject="ShowArc"/>
<objColor r="255" g="192" b="203" alpha="0"/>
<bgColor r="255" g="192" b="203" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="516" y="52"/>
<ggbscript val="SetVisibleInView[text1_1,1,false]"/>
</element>
<expression label="text3" exp=""半徑""/>
<element type="text" label="text3">
<show object="true" label="false"/>
<condition showObject="ShowArc"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="645" y="92"/>
</element>
<expression label="text3_1" exp=""半""/>
<element type="text" label="text3_1">
<show object="true" label="false"/>
<condition showObject="ShowArc"/>
<objColor r="255" g="192" b="203" alpha="0"/>
<bgColor r="255" g="192" b="203" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="640" y="92"/>
<ggbscript val="SetVisibleInView[text3_1,1,false]"/>
</element>
<expression label="text4" exp=""=""/>
<element type="text" label="text4">
<show object="true" label="false"/>
<condition showObject="ShowArc"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="709" y="92"/>
</element>
<expression label="text5" exp=""20 cm""/>
<element type="text" label="text5">
<show object="true" label="false"/>
<condition showObject="ShowArc"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="742" y="92"/>
</element>
<expression label="text5_1" exp=""20""/>
<element type="text" label="text5_1">
<show object="true" label="false"/>
<condition showObject="ShowArc"/>
<objColor r="255" g="192" b="203" alpha="0"/>
<bgColor r="255" g="192" b="203" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="742" y="92"/>
<ggbscript val="SetVisibleInView[text5_1,1,false]"/>
</element>
<expression label="text6" exp=""下圖的周界長多少？（π = 3.14）""/>
<element type="text" label="text6">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<absoluteScreenLocation x="10" y="31"/>
</element>
<expression label="text7" exp=""周界長：""/>
<element type="text" label="text7">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<absoluteScreenLocation x="11" y="428"/>
</element>
<element type="boolean" label="ShowArcMajor">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="Semicircle">
<input a0="(0, 0)" a1="(0, 2)"/>
<output a0="p"/>
</command>
<element type="conicpart" label="p">
<show object="true" label="false"/>
<condition showObject="ShowSemicircle ∧ ShowSemicircle2"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<ggbscript val="SetValue[ShowSemicircle2, !ShowSemicircle2]"/>
<lineStyle thickness="3" type="10" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="0" A3="0" A4="0" A5="-1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="ShowSemicircle2">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="text8" exp=""\frac{1}{2}""/>
<element type="text" label="text8">
<show object="true" label="false"/>
<condition showObject="ShowSemicircle"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="516" y="132"/>
</element>
<expression label="text9" exp=""個圓周；""/>
<element type="text" label="text9">
<show object="true" label="false"/>
<condition showObject="ShowSemicircle"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="544" y="172"/>
</element>
<expression label="text10" exp=""直徑""/>
<element type="text" label="text10">
<show object="true" label="false"/>
<condition showObject="ShowSemicircle"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="645" y="172"/>
</element>
<expression label="text11" exp=""=""/>
<element type="text" label="text11">
<show object="true" label="false"/>
<condition showObject="ShowSemicircle"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="709" y="172"/>
</element>
<expression label="text12" exp=""20 cm""/>
<element type="text" label="text12">
<show object="true" label="false"/>
<condition showObject="ShowSemicircle"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="742" y="172"/>
</element>
<expression label="text8_1" exp=""\frac{1}{2}""/>
<element type="text" label="text8_1">
<show object="true" label="false"/>
<condition showObject="ShowSemicircle"/>
<objColor r="255" g="192" b="203" alpha="0"/>
<bgColor r="255" g="192" b="203" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="516" y="132"/>
<ggbscript val="SetVisibleInView[text8_1,1,false]"/>
</element>
<expression label="text10_1" exp=""直""/>
<element type="text" label="text10_1">
<show object="true" label="false"/>
<condition showObject="ShowSemicircle"/>
<objColor r="255" g="192" b="203" alpha="0"/>
<bgColor r="255" g="192" b="203" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="641" y="172"/>
<ggbscript val="SetVisibleInView[text10_1,1,false]"/>
</element>
<expression label="text12_1" exp=""20""/>
<element type="text" label="text12_1">
<show object="true" label="false"/>
<condition showObject="ShowSemicircle"/>
<objColor r="255" g="192" b="203" alpha="0"/>
<bgColor r="255" g="192" b="203" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="742" y="172"/>
<ggbscript val="SetVisibleInView[text12_1,1,false]"/>
</element>
<expression label="text13" exp=""1條直線長 20 cm""/>
<element type="text" label="text13">
<show object="true" label="true"/>
<condition showObject="ShowSegment"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="516" y="253"/>
</element>
<expression label="text14" exp=""點擊曲線和直線，顯示／隱葳著色及提示。""/>
<element type="text" label="text14">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="0.7" size="-8" style="1"/>
<absoluteScreenLocation x="515" y="26"/>
</element>
</construction>
</geogebra>