Arwynebedd: Gwahaniaeth rhwng fersiynau

Cynnwys wedi'i ddileu Cynnwys wedi'i ychwanegu
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Tagiau: Golygiad cod 2017
Dim crynodeb golygu
Tagiau: Golygiad cod 2017
Llinell 10:
Mae yna sawl [[fformiwla]] adnabyddus ar gyfer arwynebedd siapiau syml fel [[Triongl|trionglau]], [[Petryal|petryalau]], a [[Cylch|chylchoedd]]. Gan ddefnyddio'r fformwlâu hyn, gellir dod o hyd i arwynebedd unrhyw [[Polygon|bolygon]] trwy rannu'r polygon yn drionglau.<ref name="bkos">{{Cite book|last=Mark de Berg|last2=Marc van Kreveld|last3=Mark Overmars|author-link3=Mark Overmars|last4=Otfried Schwarzkopf|year=2000|title=Computational Geometry|publisher=[[Springer-Verlag]]|edition=2nd revised|isbn=978-3-540-65620-3|chapter=Chapter 3: Polygon Triangulation|pages=[https://archive.org/details/computationalgeo00berg/page/45 45–61]}}</ref> Ar gyfer siap â ffin grom, fel rheol mae angen [[calcwlws]] i gyfrifo'r arwynebedd. Yn wir, roedd y broblem o bennu arwynebedd ffigurau plân yn gymhelliant mawr i ddatblygiad hanesyddol calcwlws.<ref>{{Cite book|first=Carl B.|last=Boyer|author-link=Carl Benjamin Boyer|title=A History of the Calculus and Its Conceptual Development|publisher=Dover|year=1959|isbn=978-0-486-60509-8|url=https://archive.org/details/historyofcalculu00boye}}</ref>
 
Ar gyfer siâp solet fel [[sffêr]], [[côn]], neu [[silindr]], gelwir arwynebedd ei ffin yn "[[arwyneb yr arwynebedd]]".<ref name="MathWorld 2">{{Cite web|url=http://mathworld.wolfram.com/Area.html|title=Area|publisher=[[Wolfram MathWorld]]|authorlink=Eric W. Weisstein|last=Weisstein, Eric W.|access-date=3 July 2012|archiveurl=https://web.archive.org/web/20120505085753/http://mathworld.wolfram.com/Area.html|archivedate=5 May 2012}}<cite class="citation web cs1" data-ve-ignore="true" id="CITEREFWeisstein,_Eric_W.">[[Eric W. Weisstein|Weisstein, Eric W.]] [http://mathworld.wolfram.com/Area.html "Area"]. [[Wolfram MathWorld]]. [https://web.archive.org/web/20120505085753/http://mathworld.wolfram.com/Area.html Archived] from the original on 5 May 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">3 July</span> 2012</span>.</cite></ref><ref name="MathWorldSurfaceArea">{{Cite web|url=http://mathworld.wolfram.com/SurfaceArea.html|title=Surface Area|publisher=[[Wolfram MathWorld]]|authorlink=Eric W. Weisstein|last=Weisstein, Eric W.|access-date=3 July 2012|archiveurl=https://web.archive.org/web/20120623021029/http://mathworld.wolfram.com/SurfaceArea.html|archivedate=23 June 2012}}</ref><ref>{{Cite web|url=https://www.ck12.org/c/geometry/surface-area/lesson/Surface-Area-and-Nets-GEO-CCSS/|title=Surface Area|last=Foundation|first=CK-12|website=CK-12 Foundation|language=en|access-date=2018-10-09}}</ref> Cyfrifwyd [[Fformiwla|fformiwlâu]] ar gyfer arwynebedd siapiau syml gan yr [[hen Roegiaid]], ond fel rheol mae cyfrifo arwynebedd siâp mwy cymhleth yn gofyn am galcwlws aml-newidyn (<nowiki><i>multivariable</i></nowiki>).
 
Mae arwynebedd yn chwarae rhan bwysig mewn mathemateg fodern. Yn ychwanegol at ei bwysigrwydd amlwg mewn [[geometreg]] a [[Calcwlws|chalcwlws]], mae arwynebedd yn gysylltiedig â'r diffiniad o benderfynyddion mewn [[algebra llinol]], ac mae'n briodwedd sylfaenol arwynebau mewn [[geometreg gwahaniaethol|geometreg wahaniaethol]]. Mewn dadansoddiad, diffinnir arwynebedd is-set o'r blanau gan ddefnyddio mesur Lebesgue,<ref name="Rudin">Walter Rudin (1966). ''Real and Complex Analysis'', McGraw-Hill, {{ISBN|0-07-100276-6}}.</ref> er nad yw pob is-set yn fesuradwy.<ref>Gerald Folland (1999). ''Real Analysis: modern techniques and their applications'', John Wiley & Sons, Inc., p. 20, {{ISBN|0-471-31716-0}}</ref> Yn gyffredinol, mae arwynebedd mewn mathemateg-uwch yn cael ei ystyried yn achos arbennig o gyfaint ar gyfer rhanbarthau dau-ddimensiwn. <ref name="MathWorld">{{Cite web|url=http://mathworld.wolfram.com/Area.html|title=Area|publisher=[[Wolfram MathWorld]]|authorlink=Eric W. Weisstein|last=Weisstein, Eric W.|access-date=3 July 2012|archiveurl=https://web.archive.org/web/20120505085753/http://mathworld.wolfram.com/Area.html|archivedate=5 May 2012}}<cite class="citation web cs1" data-ve-ignore="true" id="CITEREFWeisstein,_Eric_W.">[[Eric W. Weisstein|Weisstein, Eric W.]] [http://mathworld.wolfram.com/Area.html "Area"]. [[Wolfram MathWorld]]. [https://web.archive.org/web/20120505085753/http://mathworld.wolfram.com/Area.html Archived] from the original on 5 May 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">3 July</span> 2012</span>.</cite></ref>
Llinell 181:
: {{Bigmath|''A'' {{=}} ''lw''}} &nbsp;(petryal).
 
Hynny yw, arwynebedd y petryal yw'r hyd wedi'i luosi â'r lled. Fel achos arbennig, fel {{math|''l'' {{=}} ''w''}} yn achos sgwâr, rhoddir arwynebedd sgwâr â hyd ochr ''s'' yn y fformiwla:<ref name="MathWorld 3">{{Cite web|url=http://mathworld.wolfram.com/Area.html|title=Area|publisher=[[Wolfram MathWorld]]|authorlink=Eric W. Weisstein|last=Weisstein, Eric W.|access-date=3 July 2012|archiveurl=https://web.archive.org/web/20120505085753/http://mathworld.wolfram.com/Area.html|archivedate=5 May 2012}}<cite class="citation web cs1" data-ve-ignore="true" id="CITEREFWeisstein,_Eric_W.">[[Eric W. Weisstein|Weisstein, Eric W.]] [http://mathworld.wolfram.com/Area.html "Area"]. [[Wolfram MathWorld]]. [https://web.archive.org/web/20120505085753/http://mathworld.wolfram.com/Area.html Archived] from the original on 5 May 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">3 July</span> 2012</span>.</cite></ref><ref name="AF">{{Cite web|url=http://www.math.com/tables/geometry/areas.htm|title=Area Formulas|publisher=Math.com|access-date=2 July 2012|archiveurl=https://web.archive.org/web/20120702135710/http://www.math.com/tables/geometry/areas.htm|archivedate=2 July 2012}}<cite class="citation web cs1" data-ve-ignore="true">[http://www.math.com/tables/geometry/areas.htm "Area Formulas"]. Math.com. [https://web.archive.org/web/20120702135710/http://www.math.com/tables/geometry/areas.htm Archived] from the original on 2 July 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">2 July</span> 2012</span>.</cite></ref><ref>{{Cite web|url=http://proofwiki.org/wiki/Area_of_Square|title=Area of Square|publisher=ProofWiki.org|access-date=29 May 2016|archiveurl=https://web.archive.org/web/20171104234553/https://proofwiki.org/wiki/Area_of_Square|archivedate=4 November 2017}}</ref>
 
: {{Bigmath|''A'' {{=}} ''s''<sup>2</sup>}} &nbsp;(sgwâr).