What-is-the-area-of-overlapping-region-of-three-circles-of-radii-r-1-r-2-r-3-with-their-respective-centres-C-1-C-2-and-C-3-when-r-1-r-2-gt-C-1-C-2-r-2-r-3-gt-C-2-C-3-and-r-

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Question Number 3943 by Rasheed Soomro last updated on 25/Dec/15

$$\mathcal{W}hatistheareaofoverlappingregion$$$$ofthreecirclesofradii{\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_{3}withtheir$$$$respectivecentres{\mathbf{C}}_1,{\mathbf{C}}_{2}and{\mathbf{C}}_{3}when$$$${\mathbf{r}}_1+{\mathbf{r}}_2>{\mathbf{C}}_1{\mathbf{C}}_2,{\mathbf{r}}_2+{\mathbf{r}}_3>{\mathbf{C}}_2{\mathbf{C}}_{3}and$$$${\mathbf{r}}_3+{\mathbf{r}}_1>{\mathbf{C}}_3{\mathbf{C}}_1.$$$$Notethat{\mathbf{C}}_{\mathbf{i}}{\mathbf{C}}_{\mathbf{j}}isthedistancebetweencentres$$$${\mathbf{C}}_{\mathbf{i}}and{\mathbf{C}}_{\mathbf{j}}.$$

Commented by Yozzii last updated on 26/Dec/15

Commented by Yozzii last updated on 26/Dec/15

$$Yes.Iagree.Lookatmyimage$$$$forabetterideaonfindingthefull$$$$area.$$

Commented by Rasheed Soomro last updated on 26/Dec/15

$$Buttheproblemistodeterminethecentralangles$$$$ofthesearcs!$$

Commented by Yozzii last updated on 25/Dec/15

$$Itispossiblethatonecirclecompletely$$$$containstheothertwooverlapping$$$$circles.Theproblemthenrequires$$$$findingonlytheareainwhichthe$$$$twoinnercirclesoverlapsincethis$$$$samearealieswithinthelarger$$$$circle.Theconditionson{r}_{i}and{C}_i{C}_j$$$$arestillsatisfied.$$$$$$

Commented by Yozzii last updated on 25/Dec/15

$$IthinkIhaveameansoffindingtherequired$$$$areaifnotwocirclesfullycontaineach$$$$other.Itrequiresformingthreenew$$$$circlesandatriangle.Adiagram$$$$wouldtrulyhelpmyexplanation.$$$$$$$$Supposenotwocirclesfullycontain$$$$eachotherandthattheyappeartoform$$$$eightdistinctregionsintheplane,$$$$withthecommonregionofoverlaphavingthree$$$$verticesA,B,C.Constructthetriangle$$$$A,B,C.Calltheareaof△ABC{\mathit{A}}_1.$$$$WehaveAB,BCandCAasarcs,$$$$becauseofthemannerinwhichthe$$$$threecirvlesoverlap,and$$$$wecanformcirclestoadhereto$$$$eachofthesearcs.Thus,ifthisisdone$$$$wecanfindtherespectiveareas{\mathit{A}}_2,$$$${\mathit{A}}_{3}and{\mathit{A}}_4,thatliebetweeneachside$$$$ofthetriangleandthearcopposite$$$$eachside.Foragivensidewithanarc,$$$$theareadifferencebetweenthesector$$$$ofthecirclewitharcAB,fore.g,$$$$andtheisoscelestriangle{ABO}_1,$$$$where{O}_{1}isthecentreofthisnew$$$$circle,is{\mathit{A}}_2,forexample.Theareaof$$$$theregionofoverlapisthen$$$${\mathit{A}}_1+{\mathit{A}}_2+{\mathit{A}}_3+{\mathit{A}}_4.$$$$$$

Commented by Rasheed Soomro last updated on 25/Dec/15

$$Allpossiblecasesshouldbeconsidered$$$$separately$$

Commented by Yozzii last updated on 25/Dec/15

$$Ijustrealizedthatyouonlyneedthe$$$$threeinitialcirclestodothe$$$$calculation.ThenewcirclesIspeakof$$$$onlyneedtobetheinitialcircleswe$$$$startedwith!(Wasteofenergy\dots )$$

Commented by Rasheed Soomro last updated on 25/Dec/15

$$\mathcal{T}hreecoplanercirclescanformatmost$$$$sevendistinctregions.Ithink.$$

Commented by Yozzii last updated on 25/Dec/15

$$Theregionoutsideallofthemmakeit8.$$

Commented by Rasheed Soomro last updated on 26/Dec/15

$$Youshouldstartwith7-region$$$$case,Ithink.$$$$Ialsothinkthattheareaofoverlappingregionisclosely/direcly$$$$relatedtothequantities:$$$${\mathbf{r}}_1+{\mathbf{r}}_2-{\mathbf{C}}_1{\mathbf{C}}_2,{\mathbf{r}}_2+{\mathbf{r}}_3-{\mathbf{C}}_2{\mathbf{C}}_3,{\mathbf{r}}_3+{\mathbf{r}}_1-{\mathbf{C}}_3{\mathbf{C}}_1$$$$Morethecirclesareclosedgreaterthearea$$$$ofoverlappingregion.$$

Commented by Yozzii last updated on 26/Dec/15

$$A,C\mathrm{\&}Earecirclecentres.Tofind$$$$theareabetweenthearcHIandthe$$$$chordHI,findtheareaof△CHI$$$$andthentheareaofthesector$$$$CHI.Thepositivedifferencebetween$$$$thetwoareasgiveswhat{we}^{\prime }re$$$$searchingfor.Thisissimilarly$$$$doneforsectorEGHandsectorAGI.$$$$$$

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