Confusion on Parallelogram












1












$begingroup$


i know it's rude and pretentious and maybe stupid to ask such question but, i would like to know the reason why in geometry the area of a parallelogram follows this formula :



$ Base times Height $



and not this one :



$ Base times Side $



I saw this method which consists of transforming our figure into a rectangle by moving the black one to the right, and that seems logical to me, but why not adopt the second rule, all the horizontal points multiplied by the vertical ones ?



enter image description here



PS : I found the same Visualizing the Area of a Parallelogram, but was unconvinced by the answers






geometry







share|cite|improve this question









New contributor




Bo Halim is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 2




    $begingroup$
    Possible duplicate of Visualizing the Area of a Parallelogram
    $endgroup$
    – kkc
    7 hours ago










  • $begingroup$
    @kkc, did you see the link ?
    $endgroup$
    – Bo Halim
    7 hours ago










  • $begingroup$
    In mathematics, there is nothing rude and pretentious about asking questions
    $endgroup$
    – Taladris
    21 mins ago
















1












$begingroup$


i know it's rude and pretentious and maybe stupid to ask such question but, i would like to know the reason why in geometry the area of a parallelogram follows this formula :



$ Base times Height $



and not this one :



$ Base times Side $



I saw this method which consists of transforming our figure into a rectangle by moving the black one to the right, and that seems logical to me, but why not adopt the second rule, all the horizontal points multiplied by the vertical ones ?



enter image description here



PS : I found the same Visualizing the Area of a Parallelogram, but was unconvinced by the answers






geometry







share|cite|improve this question









New contributor




Bo Halim is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 2




    $begingroup$
    Possible duplicate of Visualizing the Area of a Parallelogram
    $endgroup$
    – kkc
    7 hours ago










  • $begingroup$
    @kkc, did you see the link ?
    $endgroup$
    – Bo Halim
    7 hours ago










  • $begingroup$
    In mathematics, there is nothing rude and pretentious about asking questions
    $endgroup$
    – Taladris
    21 mins ago














1












1








1





$begingroup$


i know it's rude and pretentious and maybe stupid to ask such question but, i would like to know the reason why in geometry the area of a parallelogram follows this formula :



$ Base times Height $



and not this one :



$ Base times Side $



I saw this method which consists of transforming our figure into a rectangle by moving the black one to the right, and that seems logical to me, but why not adopt the second rule, all the horizontal points multiplied by the vertical ones ?



enter image description here



PS : I found the same Visualizing the Area of a Parallelogram, but was unconvinced by the answers






geometry







share|cite|improve this question









New contributor




Bo Halim is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




i know it's rude and pretentious and maybe stupid to ask such question but, i would like to know the reason why in geometry the area of a parallelogram follows this formula :



$ Base times Height $



and not this one :



$ Base times Side $



I saw this method which consists of transforming our figure into a rectangle by moving the black one to the right, and that seems logical to me, but why not adopt the second rule, all the horizontal points multiplied by the vertical ones ?



enter image description here



PS : I found the same Visualizing the Area of a Parallelogram, but was unconvinced by the answers






geometry




geometry






share|cite|improve this question









New contributor




Bo Halim is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Bo Halim is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 2 hours ago









dantopa

6,63342245




6,63342245






New contributor




Bo Halim is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 7 hours ago









Bo HalimBo Halim

1094




1094




New contributor




Bo Halim is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Bo Halim is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Bo Halim is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 2




    $begingroup$
    Possible duplicate of Visualizing the Area of a Parallelogram
    $endgroup$
    – kkc
    7 hours ago










  • $begingroup$
    @kkc, did you see the link ?
    $endgroup$
    – Bo Halim
    7 hours ago










  • $begingroup$
    In mathematics, there is nothing rude and pretentious about asking questions
    $endgroup$
    – Taladris
    21 mins ago














  • 2




    $begingroup$
    Possible duplicate of Visualizing the Area of a Parallelogram
    $endgroup$
    – kkc
    7 hours ago










  • $begingroup$
    @kkc, did you see the link ?
    $endgroup$
    – Bo Halim
    7 hours ago










  • $begingroup$
    In mathematics, there is nothing rude and pretentious about asking questions
    $endgroup$
    – Taladris
    21 mins ago








2




2




$begingroup$
Possible duplicate of Visualizing the Area of a Parallelogram
$endgroup$
– kkc
7 hours ago




$begingroup$
Possible duplicate of Visualizing the Area of a Parallelogram
$endgroup$
– kkc
7 hours ago












$begingroup$
@kkc, did you see the link ?
$endgroup$
– Bo Halim
7 hours ago




$begingroup$
@kkc, did you see the link ?
$endgroup$
– Bo Halim
7 hours ago












$begingroup$
In mathematics, there is nothing rude and pretentious about asking questions
$endgroup$
– Taladris
21 mins ago




$begingroup$
In mathematics, there is nothing rude and pretentious about asking questions
$endgroup$
– Taladris
21 mins ago










3 Answers
3






active

oldest

votes


















7












$begingroup$

Sometimes a figure is worth 1000 words:



parallelogram



Very long base and very long side and very small area.



Or...



...each of these parallelograms has the same base and side, but manifestly different areas:



enter image description here






share|cite|improve this answer











$endgroup$





















    4












    $begingroup$

    You can't "adopt the second rule" because it gives the wrong answer. The parallelogram in your picture has all four sides of length $5$, There are lots of parallelograms like that, depending on the angle between adjacent sides. The largest is a square. If you make the acute angle smaller and smaller you find parallelograms that are nearly on a line, with tiny areas. But your formula would say they all have area $25$.






    share|cite|improve this answer









    $endgroup$





















      1












      $begingroup$

      As you point out, the first method is logical, as it uses a formula we already know to be true (the familiar area formula for a rectangle), and then transforms our figure without changing the area, so that we can apply the rule.



      To see why we can't simply multiply $5$ and $5$ to get the answer, draw the $5times 5$ square. Your parallelogram can't fill it by cut and paste. Alternatively, you can draw $25$ unit squares, and try to cut and paste them into the parallelogram. You'll only be able to make $20$ of them fit.






      share|cite|improve this answer









      $endgroup$













        Your Answer





        StackExchange.ifUsing("editor", function () {
        return StackExchange.using("mathjaxEditing", function () {
        StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
        StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
        });
        });
        }, "mathjax-editing");

        StackExchange.ready(function() {
        var channelOptions = {
        tags: "".split(" "),
        id: "69"
        };
        initTagRenderer("".split(" "), "".split(" "), channelOptions);

        StackExchange.using("externalEditor", function() {
        // Have to fire editor after snippets, if snippets enabled
        if (StackExchange.settings.snippets.snippetsEnabled) {
        StackExchange.using("snippets", function() {
        createEditor();
        });
        }
        else {
        createEditor();
        }
        });

        function createEditor() {
        StackExchange.prepareEditor({
        heartbeatType: 'answer',
        autoActivateHeartbeat: false,
        convertImagesToLinks: true,
        noModals: true,
        showLowRepImageUploadWarning: true,
        reputationToPostImages: 10,
        bindNavPrevention: true,
        postfix: "",
        imageUploader: {
        brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
        contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
        allowUrls: true
        },
        noCode: true, onDemand: true,
        discardSelector: ".discard-answer"
        ,immediatelyShowMarkdownHelp:true
        });


        }
        });






        Bo Halim is a new contributor. Be nice, and check out our Code of Conduct.










        draft saved

        draft discarded


















        StackExchange.ready(
        function () {
        StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3161100%2fconfusion-on-parallelogram%23new-answer', 'question_page');
        }
        );

        Post as a guest















        Required, but never shown

























        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        7












        $begingroup$

        Sometimes a figure is worth 1000 words:



        parallelogram



        Very long base and very long side and very small area.



        Or...



        ...each of these parallelograms has the same base and side, but manifestly different areas:



        enter image description here






        share|cite|improve this answer











        $endgroup$


















          7












          $begingroup$

          Sometimes a figure is worth 1000 words:



          parallelogram



          Very long base and very long side and very small area.



          Or...



          ...each of these parallelograms has the same base and side, but manifestly different areas:



          enter image description here






          share|cite|improve this answer











          $endgroup$
















            7












            7








            7





            $begingroup$

            Sometimes a figure is worth 1000 words:



            parallelogram



            Very long base and very long side and very small area.



            Or...



            ...each of these parallelograms has the same base and side, but manifestly different areas:



            enter image description here






            share|cite|improve this answer











            $endgroup$



            Sometimes a figure is worth 1000 words:



            parallelogram



            Very long base and very long side and very small area.



            Or...



            ...each of these parallelograms has the same base and side, but manifestly different areas:



            enter image description here







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 6 hours ago

























            answered 7 hours ago









            David G. StorkDavid G. Stork

            11.3k41432




            11.3k41432























                4












                $begingroup$

                You can't "adopt the second rule" because it gives the wrong answer. The parallelogram in your picture has all four sides of length $5$, There are lots of parallelograms like that, depending on the angle between adjacent sides. The largest is a square. If you make the acute angle smaller and smaller you find parallelograms that are nearly on a line, with tiny areas. But your formula would say they all have area $25$.






                share|cite|improve this answer









                $endgroup$


















                  4












                  $begingroup$

                  You can't "adopt the second rule" because it gives the wrong answer. The parallelogram in your picture has all four sides of length $5$, There are lots of parallelograms like that, depending on the angle between adjacent sides. The largest is a square. If you make the acute angle smaller and smaller you find parallelograms that are nearly on a line, with tiny areas. But your formula would say they all have area $25$.






                  share|cite|improve this answer









                  $endgroup$
















                    4












                    4








                    4





                    $begingroup$

                    You can't "adopt the second rule" because it gives the wrong answer. The parallelogram in your picture has all four sides of length $5$, There are lots of parallelograms like that, depending on the angle between adjacent sides. The largest is a square. If you make the acute angle smaller and smaller you find parallelograms that are nearly on a line, with tiny areas. But your formula would say they all have area $25$.






                    share|cite|improve this answer









                    $endgroup$



                    You can't "adopt the second rule" because it gives the wrong answer. The parallelogram in your picture has all four sides of length $5$, There are lots of parallelograms like that, depending on the angle between adjacent sides. The largest is a square. If you make the acute angle smaller and smaller you find parallelograms that are nearly on a line, with tiny areas. But your formula would say they all have area $25$.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 7 hours ago









                    Ethan BolkerEthan Bolker

                    45.3k553120




                    45.3k553120























                        1












                        $begingroup$

                        As you point out, the first method is logical, as it uses a formula we already know to be true (the familiar area formula for a rectangle), and then transforms our figure without changing the area, so that we can apply the rule.



                        To see why we can't simply multiply $5$ and $5$ to get the answer, draw the $5times 5$ square. Your parallelogram can't fill it by cut and paste. Alternatively, you can draw $25$ unit squares, and try to cut and paste them into the parallelogram. You'll only be able to make $20$ of them fit.






                        share|cite|improve this answer









                        $endgroup$


















                          1












                          $begingroup$

                          As you point out, the first method is logical, as it uses a formula we already know to be true (the familiar area formula for a rectangle), and then transforms our figure without changing the area, so that we can apply the rule.



                          To see why we can't simply multiply $5$ and $5$ to get the answer, draw the $5times 5$ square. Your parallelogram can't fill it by cut and paste. Alternatively, you can draw $25$ unit squares, and try to cut and paste them into the parallelogram. You'll only be able to make $20$ of them fit.






                          share|cite|improve this answer









                          $endgroup$
















                            1












                            1








                            1





                            $begingroup$

                            As you point out, the first method is logical, as it uses a formula we already know to be true (the familiar area formula for a rectangle), and then transforms our figure without changing the area, so that we can apply the rule.



                            To see why we can't simply multiply $5$ and $5$ to get the answer, draw the $5times 5$ square. Your parallelogram can't fill it by cut and paste. Alternatively, you can draw $25$ unit squares, and try to cut and paste them into the parallelogram. You'll only be able to make $20$ of them fit.






                            share|cite|improve this answer









                            $endgroup$



                            As you point out, the first method is logical, as it uses a formula we already know to be true (the familiar area formula for a rectangle), and then transforms our figure without changing the area, so that we can apply the rule.



                            To see why we can't simply multiply $5$ and $5$ to get the answer, draw the $5times 5$ square. Your parallelogram can't fill it by cut and paste. Alternatively, you can draw $25$ unit squares, and try to cut and paste them into the parallelogram. You'll only be able to make $20$ of them fit.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 7 hours ago









                            Cameron BuieCameron Buie

                            86.2k772161




                            86.2k772161






















                                Bo Halim is a new contributor. Be nice, and check out our Code of Conduct.










                                draft saved

                                draft discarded


















                                Bo Halim is a new contributor. Be nice, and check out our Code of Conduct.













                                Bo Halim is a new contributor. Be nice, and check out our Code of Conduct.












                                Bo Halim is a new contributor. Be nice, and check out our Code of Conduct.
















                                Thanks for contributing an answer to Mathematics Stack Exchange!


                                • Please be sure to answer the question. Provide details and share your research!

                                But avoid …



                                • Asking for help, clarification, or responding to other answers.

                                • Making statements based on opinion; back them up with references or personal experience.


                                Use MathJax to format equations. MathJax reference.


                                To learn more, see our tips on writing great answers.




                                draft saved


                                draft discarded














                                StackExchange.ready(
                                function () {
                                StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3161100%2fconfusion-on-parallelogram%23new-answer', 'question_page');
                                }
                                );

                                Post as a guest















                                Required, but never shown





















































                                Required, but never shown














                                Required, but never shown












                                Required, but never shown







                                Required, but never shown

































                                Required, but never shown














                                Required, but never shown












                                Required, but never shown







                                Required, but never shown







                                -

                                Popular posts from this blog

                                Bundesstraße 106

                                Image
                                Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück
                                Read more

                                Verónica Boquete

                                Image
                                Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl
                                Read more

                                Ida-Boy-Ed-Garten

                                Image
                                Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich
                                Read more