Area of a Parallelogram: Derivation and Demonstration
2013-05-06 10:09
This GeoGebra applet demonstrates the area formula of a parallelogram. Using geometric constructions and transformations like translation and rotation, it converts a parallelogram into a rectangle of equal area, illustrating the base times height principle. It features dynamic measurements and interactive elements, making it an ideal tool for geometry education.
In collections
Polygons: Interactive GeoGebra Geometry Book
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.2" version="4.2.31.0" id="3ef23f43-4ee6-4642-bc05-1ddd44f6f1a1" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1288" height="1004" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.19875776397515527" orientation="1" />
</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="1,1" size="200" window="0,0,400,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1280" window="0,0,400,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" 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="379,361,500,500" />
<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="false" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="1280" height="874"/>
<coordSystem xZero="215.0" yZero="315.0" scale="30.000000000000007" yscale="30.0"/>
<evSettings axes="false" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="2" checkboxSize="13" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="64" g="64" b="64"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="10"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="0" showNumbers="true" tickDistance="1.0"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="0" showNumbers="true" tickDistance="1.0"/>
<grid distX="1.0" distY="1.0" distTheta="0.5235987755982988"/>
</euclidianView>
<kernel>
<continuous val="true"/>
<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="Area of a parallelogram" author="Duane Habecker" date="6/16/07">
<worksheetText above="DIRECTIONS: 1. Move the blue vertices and lines b and c to create the parallelogram of your choice. 2. Pull the red slider to observe the animation. 3. How does the area of the original parallelogram compare to the area of the newly created rectangle? " below=""/>
<element type="line" label="c">
<show object="true" label="true"/>
<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="-2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="explicit"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-1" y="24"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="5.0" y="2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="line" label="b">
<show object="true" label="true"/>
<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="-7.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="explicit"/>
</element>
<command name="Point">
<input a0="b"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-14" y="-4"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-2.0" y="7.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="D" a1="c"/>
<output a0="d"/>
</command>
<element type="line" label="d">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="0.0" z="-2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="c" a1="d"/>
<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"/>
<labelOffset x="-5" y="22"/>
<labelMode val="0"/>
<coords x="-2.0" y="2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-7" y="34"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-4.0" y="2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="7.8" y="2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="B" a1="A" a2="D"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<value val="1.1902899496825317"/>
<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>
<command name="Rotate">
<input a0="C" a1="α" a2="B"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.039893893791491" y="4.5997347344787265" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="C" a1="B" a2="E"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<value val="1.1902899496825317"/>
<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>
<command name="Line">
<input a0="B" a1="E"/>
<output a0="a"/>
</command>
<element type="line" label="a">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.5997347344787265" y="1.0398938937914908" z="10.918885884810651"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="b" a1="a"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="false" label="false"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="18.198143141351085" y="18.198143141351085" z="2.5997347344787265"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="D" a1="A" a2="G"/>
<output a0="P" a1="g" a2="d_1" a3="a_1"/>
</command>
<element type="polygon" label="P">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="g">
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.0" y="-2.0" z="24.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d_1">
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="2.0" z="-4.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.0" y="0.0" z="-10.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="G" a1="B" a2="F" a3="D"/>
<output a0="Q" a1="g_1" a2="b_1" a3="底" a4="d_2"/>
</command>
<element type="polygon" label="Q">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="g_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="7.0" z="-14.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.0" y="2.0" z="21.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="底">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-29" y="8"/>
<labelMode val="1"/>
<coords x="0.0" y="-9.0" z="63.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d_2">
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.0" y="0.0" z="10.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Distance">
<input a0="g" a1="g_1"/>
<output a0="h"/>
</command>
<element type="numeric" label="h">
<value val="0.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<element type="numeric" label="e">
<value val="100.0"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="100" absoluteScreenLocation="true" width="100.0" x="185.0" y="285.0" fixed="true" horizontal="true"/>
<lineStyle thickness="5" type="10" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="ttt" exp="1 - (100 - (100 - e)) / 100" />
<element type="numeric" label="ttt">
<value val="0.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Vector">
<input a0="((d_1 + g_1) * (1 - ttt), 0)"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.0" y="0.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<command name="Translate">
<input a0="D" a1="u"/>
<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="0"/>
<labelMode val="0"/>
<coords x="7.0" y="7.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Translate">
<input a0="A" a1="u"/>
<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="0"/>
<labelMode val="0"/>
<coords x="5.0" y="2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Translate">
<input a0="G" a1="u"/>
<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="7.0" y="2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="D'" a1="A'" a2="G'"/>
<output a0="P'" a1="g'" a2="d'" a3="高"/>
</command>
<element type="polygon" label="P'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="g'">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.0" y="-2.0" z="-21.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d'">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="2.0" z="-4.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="高">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-9" y="6"/>
<labelMode val="1"/>
<coords x="-5.0" y="0.0" z="35.0"/>
<lineStyle thickness="3" type="10" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="T1" exp=""Pull slider to see animation.""/>
<element type="text" label="T1">
<show object="false" label="true"/>
<objColor r="255" g="0" b="204" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="0.8333333333333334" size="-2" style="3"/>
<absoluteScreenLocation x="165" y="306"/>
</element>
<element type="point" label="AnchorPoint">
<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="-3.0" y="-1.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="If">
<input a0="e ≟ 100" a1="AnchorPoint"/>
<output a0="Showanswer"/>
</command>
<element type="point" label="Showanswer">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.0" y="-1.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="T3" exp=""長方形面積的公式是：長 x 闊 同學，你可以猜到平行四邊形面積的公式是什麼嗎﹖""/>
<element type="text" label="T3">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<startPoint exp="Showanswer"/>
<labelOffset x="4" y="80"/>
</element>
<expression label="text1" exp=""分割圖形""/>
<element type="text" label="text1">
<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="1.4" size="5" style="0"/>
<startPoint x="-1.0" y="0.0" z="1.0"/>
</element>
<expression label="T2" exp=""長方形面積\,\,=\,\,底\,x\,高\,\,=\,\," + (LaTeX[底]) + "\,x\," + (LaTeX[高]) + "\,\,=\,\," + (LaTeX[底 * 高]) + "\,cm^2"" />
<element type="text" label="T2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.5" size="6" style="0"/>
<startPoint x="-2.999999999999999" y="-1.0" z="1.0"/>
</element>
</construction>
</geogebra>