Question Number 3943 by Rasheed Soomro last updated on 25/Dec/15
$$\mathcal{W}{hat}\:{is}\:{the}\:{area}\:\:{of}\:\:{overlapping}\:{region} \\ $$$${of}\:{three}\:{circles}\:{of}\:{radii}\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{3}} \:{with}\:{their} \\ $$$${respective}\:{centres}\:\boldsymbol{\mathrm{C}}_{\mathrm{1}} \:,\:\boldsymbol{\mathrm{C}}_{\mathrm{2}} \:{and}\:\boldsymbol{\mathrm{C}}_{\mathrm{3}} \:{when} \\ $$$$\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{1}} \boldsymbol{\mathrm{C}}_{\mathrm{2}} \:\:,\:\:\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} +\boldsymbol{\mathrm{r}}_{\mathrm{3}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{2}} \boldsymbol{\mathrm{C}}_{\mathrm{3}} \:\:{and} \\ $$$$\boldsymbol{\mathrm{r}}_{\mathrm{3}} +\boldsymbol{\mathrm{r}}_{\mathrm{1}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{3}} \boldsymbol{\mathrm{C}}_{\mathrm{1}} . \\ $$$${Note}\:{that}\:\boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{i}}} \boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{j}}} \:{is}\:{the}\:{distance}\:{between}\:{centres} \\ $$$$\boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{i}}\:} {and}\:\:\boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{j}}} \:. \\ $$
Commented by Yozzii last updated on 26/Dec/15
Commented by Yozzii last updated on 26/Dec/15
$${Yes}.\:{I}\:{agree}.\:{Look}\:{at}\:{my}\:{image} \\ $$$${for}\:{a}\:{better}\:{idea}\:{on}\:{finding}\:{the}\:{full} \\ $$$${area}. \\ $$
Commented by Rasheed Soomro last updated on 26/Dec/15
$${But}\:{the}\:{problem}\:{is}\:{to}\:{determine}\:{the}\:{central}\:{angles} \\ $$$${of}\:{these}\:{arcs}! \\ $$
Commented by Yozzii last updated on 25/Dec/15
$${It}\:{is}\:{possible}\:{that}\:{one}\:{circle}\:{completely} \\ $$$${contains}\:{the}\:{other}\:{two}\:{overlapping} \\ $$$${circles}.\:{The}\:{problem}\:{then}\:{requires} \\ $$$${finding}\:{only}\:{the}\:{area}\:{in}\:{which}\:{the} \\ $$$${two}\:{inner}\:{circles}\:{overlap}\:{since}\:{this} \\ $$$${same}\:{area}\:{lies}\:{within}\:{the}\:{larger}\: \\ $$$${circle}.\:{The}\:{conditions}\:{on}\:{r}_{{i}} \:{and}\:{C}_{{i}} {C}_{{j}} \\ $$$${are}\:{still}\:{satisfied}.\: \\ $$$$ \\ $$
Commented by Yozzii last updated on 25/Dec/15
$${I}\:{think}\:{I}\:{have}\:{a}\:{means}\:{of}\:{finding}\:{the}\:{required} \\ $$$${area}\:{if}\:{no}\:{two}\:{circles}\:{fully}\:{contain}\:{each}\: \\ $$$${other}.\:{It}\:{requires}\:{forming}\:{three}\:{new} \\ $$$${circles}\:{and}\:{a}\:{triangle}.\:{A}\:{diagram}\: \\ $$$${would}\:{truly}\:{help}\:{my}\:{explanation}.\: \\ $$$$ \\ $$$${Suppose}\:{no}\:{two}\:{circles}\:{fully}\:{contain} \\ $$$${each}\:{other}\:{and}\:{that}\:{they}\:{appear}\:{to}\:{form} \\ $$$${eight}\:{distinct}\:{regions}\:{in}\:{the}\:{plane}, \\ $$$${with}\:{the}\:{common}\:{region}\:{of}\:{overlap}\:{having}\:{three} \\ $$$${vertices}\:{A},{B},{C}.\:{Construct}\:{the}\:{triangle} \\ $$$${A},{B},{C}.\:{Call}\:{the}\:{area}\:{of}\:\bigtriangleup{ABC}\:\boldsymbol{{A}}_{\mathrm{1}} . \\ $$$${We}\:{have}\:{AB},{BC}\:{and}\:{CA}\:{as}\:{arcs}, \\ $$$${because}\:{of}\:{the}\:{manner}\:{in}\:{which}\:{the} \\ $$$${three}\:{cirvles}\:{overlap},\:{and} \\ $$$$\:{we}\:{can}\:{form}\:{circles}\:{to}\:{adhere}\:{to} \\ $$$${each}\:{of}\:{these}\:{arcs}.\:{Thus},\:{if}\:{this}\:{is}\:{done} \\ $$$${we}\:{can}\:{find}\:{the}\:{respective}\:{areas}\:\boldsymbol{{A}}_{\mathrm{2}} , \\ $$$$\boldsymbol{{A}}_{\mathrm{3}} \:{and}\:\boldsymbol{{A}}_{\mathrm{4}} ,\:{that}\:{lie}\:{between}\:{each}\:{side} \\ $$$${of}\:{the}\:{triangle}\:{and}\:{the}\:{arc}\:{opposite}\: \\ $$$${each}\:{side}.\:{For}\:{a}\:{given}\:{side}\:{with}\:{an}\:{arc}, \\ $$$${the}\:{area}\:{difference}\:{between}\:{the}\:{sector} \\ $$$${of}\:{the}\:{circle}\:{with}\:{arc}\:{AB},\:{for}\:{e}.{g},\: \\ $$$${and}\:{the}\:{isosceles}\:{triangle}\:{ABO}_{\mathrm{1}} , \\ $$$${where}\:{O}_{\mathrm{1}} \:{is}\:{the}\:{centre}\:{of}\:{this}\:{new}\: \\ $$$${circle},\:{is}\:\boldsymbol{{A}}_{\mathrm{2}} ,\:{for}\:{example}.\:{The}\:{area}\:{of} \\ $$$${the}\:{region}\:{of}\:{overlap}\:{is}\:{then}\: \\ $$$$\boldsymbol{{A}}_{\mathrm{1}} +\boldsymbol{{A}}_{\mathrm{2}} +\boldsymbol{{A}}_{\mathrm{3}} +\boldsymbol{{A}}_{\mathrm{4}} .\: \\ $$$$ \\ $$
Commented by Rasheed Soomro last updated on 25/Dec/15
$${All}\:{possible}\:{cases}\:{should}\:{be}\:{considered} \\ $$$${separately} \\ $$
Commented by Yozzii last updated on 25/Dec/15
$${I}\:{just}\:{realized}\:{that}\:{you}\:{only}\:{need}\:{the} \\ $$$${three}\:{initial}\:{circles}\:{to}\:{do}\:{the} \\ $$$${calculation}.\:{The}\:{new}\:{circles}\:{I}\:{speak}\:{of} \\ $$$${only}\:{need}\:{to}\:{be}\:{the}\:{initial}\:{circles}\:{we}\: \\ $$$${started}\:{with}!\:\left({Waste}\:{of}\:{energy}…\right) \\ $$
Commented by Rasheed Soomro last updated on 25/Dec/15
$$\mathcal{T}{hree}\:{coplaner}\:{circles}\:{can}\:{form}\:\:{atmost} \\ $$$${seven}\:{distinct}\:{regions}.\:{I}\:{think}. \\ $$
Commented by Yozzii last updated on 25/Dec/15
$${The}\:{region}\:{outside}\:{all}\:{of}\:{them}\:{make}\:{it}\:\mathrm{8}. \\ $$
Commented by Rasheed Soomro last updated on 26/Dec/15
$${You}\:\:{should}\:{start}\:{with}\:\mathrm{7}−{region}\: \\ $$$${case}\:,\:{I}\:{think}. \\ $$$${I}\:{also}\:\:{think}\:{that}\:{the}\:{area}\:{of}\:{overlapping}\:{region}\:{is}\:{closely}/{direcly} \\ $$$${related}\:{to}\:{the}\:{quantities}:\: \\ $$$$\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} −\boldsymbol{\mathrm{C}}_{\mathrm{1}} \boldsymbol{\mathrm{C}}_{\mathrm{2}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} +\boldsymbol{\mathrm{r}}_{\mathrm{3}} −\boldsymbol{\mathrm{C}}_{\mathrm{2}} \boldsymbol{\mathrm{C}}_{\mathrm{3}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{3}} +\boldsymbol{\mathrm{r}}_{\mathrm{1}} −\boldsymbol{\mathrm{C}}_{\mathrm{3}} \boldsymbol{\mathrm{C}}_{\mathrm{1}} \\ $$$${More}\:{the}\:{circles}\:{are}\:{closed}\:{greater}\:{the}\:{area}\: \\ $$$${of}\:{overlapping}\:{region}. \\ $$
Commented by Yozzii last updated on 26/Dec/15
$${A},{C}\:\&{E}\:{are}\:{circle}\:{centres}.\:{To}\:{find} \\ $$$${the}\:{area}\:{between}\:{the}\:{arc}\:{HI}\:{and}\:{the} \\ $$$${chord}\:{HI},\:{find}\:{the}\:{area}\:{of}\:\bigtriangleup{CHI} \\ $$$${and}\:{then}\:{the}\:{area}\:{of}\:{the}\:{sector} \\ $$$${CHI}.\:{The}\:{positive}\:{difference}\:{between} \\ $$$${the}\:{two}\:{areas}\:{gives}\:{what}\:{we}'{re}\: \\ $$$${searching}\:{for}.\:{This}\:{is}\:{similarly} \\ $$$${done}\:{for}\:{sector}\:{EGH}\:{and}\:{sector}\:{AGI}. \\ $$$$ \\ $$