Arwynebedd: Gwahaniaeth rhwng fersiynau
Cynnwys wedi'i ddileu Cynnwys wedi'i ychwanegu
Dim crynodeb golygu Tagiau: Golygiad cod 2017 |
Dim crynodeb golygu Tagiau: Golygiad cod 2017 |
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Llinell 188:
=== Rhestr o fformiwlâu ===
{| class="wikitable"
|+Rhestr o fformiwlâu cyffredinol ar gyfer arwynebedd:
!Siap
!Fformiwla
!Newidynnau
|-
|[[Petryal]]
|<math>A=ab</math>
|[[File:Rechteck-ab.svg|120px]]
|-
|[[Triongl]]
|<math>A=\frac12bh \,\!</math>
|[[File:Dreieck-allg-bh.svg|100px]]
|-
|[[Triongl]]
|<math>A=\frac12 a b \sin(\gamma)\,\!</math>
|[[File:Dreieck-allg-w.svg|100px]]
|-
|[[Triongl]]<br /><br />
(Fformiwla Heron)
|<math>A=\sqrt{s(s-a)(s-b)(s-c)}\,\!</math>
|[[File:Dreieck-allg.svg|100x100px]] <math> s =\tfrac 1 2 (a+b+c)</math>
|-
|[[Triongl isosgeles]]
|<math>A=\frac{b}{4}\sqrt{4a^2-c^2}</math>
|[[File:Dreieck-gsch.svg|80px]]
|-
|[[Triongl|Triongl rheolaidd]]<br /><br />
([[Triongl hafalochrog|equilateral triangle]])
|<math>A=\frac{\sqrt{3}}{4}a^2\,\!</math>
||[[File:Dreieck-gseit.svg|100px]]
|-
|[[Rhombws]]/[[Barcut (geometreg)|Barcud]]
|<math>A=\frac12de</math>
|[[File:Raute-de.svg|160px]]
|-
|[[Paralelogram]]
|<math>A=ah_a\,\!</math>
|[[File:Parallelog-aha.svg|160px]]
|-
|[[Trapesiwm]]
|<math>A=\frac{(a+c)h}{2} \,\!</math>
|[[File:Trapez-abcdh.svg|150px]]
|-
|[[Hecsagon|Hecsagon rheolaidd]]
|<math>A=\frac{3}{2} \sqrt{3}a^2\,\!</math>
|[[File:Hexagon-a.svg|100px]]
|-
|[[Octagon|Octagon rheolaidd]]
|<math>A=2(1+\sqrt{2})a^2\,\!</math>
|[[File:Oktagon-a.svg|120px]]
|-
|[[Polygon rheolaidd]]<br /><br />
(<math>n</math> sides)
|<math>A=n\frac{ar}{2}=\frac{pr}{2}</math><br /><br />
<math>\quad =\tfrac 1 4 na^2\cot(\tfrac \pi n)</math>
<math>\quad = nr^2 \tan(\tfrac \pi n)</math>
<math>\quad =\tfrac{1}{4n}p^2\cot(\tfrac \pi n)</math>
<math>\quad =\tfrac{1}{2}nR^2 \sin(\tfrac{2\pi}{n}) \,\!</math>
|[[Delwedd:Oktagon-a-r-R.svg|chwith|150x150px]]
<math>p=na\ </math> ([[Perimedr]])
<math>r=\tfrac a 2 \cot(\tfrac \pi n),</math>
<math>\tfrac a 2= r\tan(\tfrac \pi n)=R\sin(\tfrac \pi n)</math>
<math>r:</math> [[Mewngylch ac allgylch|mewngylch]] radiws
<math>R:</math> amgylch
|-
|[[Cylch]]
|<math>A=\pi r^2=\frac{\pi d^2}{4}</math><br /><br />
(<math> d=2r: </math> [[Diamedr]])
|[[fILE:Kreis-r-tab.svg|100px]]
|-
|Sector cylch
|<math>A=\frac{\theta}{2}r^2=\frac{L \cdot r}{2}\,\!</math>
|[[File:Circle arc.svg|120px]]
|-
|[[Elíps]]
|<math>A=\pi ab \,\!</math>
|[[File:Ellipse-ab-tab.svg|120px]]
|-
|[[Integryn]]
|<math>A=\int_a^b f(x)\mathrm{d}x ,\ f(x)\ge 0</math>
| [[File:Vase-f-fx-tab.svg|hochkant=0.2]]
|-
|
|'''Arwynebedd yr arwyneb'''
|
|-
|[[Sffêr]]<br /><br />
|<math>A = 4\pi r^2 = \pi d^2</math>
| [[File:Kugel-1-tab.svg|100px]]
|-
|[[Ciwboid]]
|<math>A = 2(ab+ac+bc)</math>
| [[File:Quader-1-tab.svg|150px]]
|-
|[[Silindr]]<br /><br />
(incl. bottom and top)
|<math>A = 2 \pi r (r + h)</math>
| [[File:Zylinder-1-tab.svg|120px]]
|-
|[[Côn]]<br /><br />
(incl. bottom)
|<math>A = \pi r (r + \sqrt{r^2+h^2})</math>
| [[File:Kegel-1-tab.svg|120px]]
|-
|Torws
|<math>A = 4\pi^2 \cdot R \cdot r</math>
| [[File:Torus-1-tab.svg|200px]]
|-
|Arwynebedd y cylchdro
|<math>A = 2\pi\int_a^b\! f(x)\sqrt{1+\left[f'(x)\right]^2}\mathrm{d}x</math><br /><br />
(cylchdro o amgylch echelin-x )
| [[File:Vase-1-tab.svg|220px]]
|-
|}
Mae'r cyfrifiadau uchod yn dangos sut i ddod o hyd i arwynebedd llawer o [[Siâp|siapiau]] cyffredin.
=== Perthynas yr arwynebedd â'r perimedr ===
Mae'r anghydraddoldeb isoperimetrig yn nodi, ar gyfer cromlin gaeedig o hyd ''L'' (felly mae gan y rhanbarth y mae'n ei amgáu [[Perimedr|berimedr]] ''L'' ) ac ar gyfer arwynebedd ''A'' y rhanbarth y mae'n ei amgáu,
: <math>4\pi A \le L^2,</math>
ac mae cydraddoldeb yn dal [[os a dim ond os]] yw'r gromlin yn [[Cylch|gylch]]. Felly, y cylch sydd â'r arwynebedd mwyaf o unrhyw ffigur caeedig gyda pherimedr penodol.
== Cyfeiriadau ==
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