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

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 > C_1 C_2 , r_2 +r_3 > C_2 C_3 and r_3 +r_1 > C_3 C_1 . Note that C_i C_j is the distance between centres C_(i ) and C_j .

$$\mathcal{W}{hat}\:{is}\:{the}\:{area}\:\:{of}\:\:{overlapping}\:{region} \\ $$ $${of}\:{three}\:{circles}\:{of}\:{radii}\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{3}} \:{with}\:{their} \\ $$ $${respective}\:{centres}\:\boldsymbol{\mathrm{C}}_{\mathrm{1}} \:,\:\boldsymbol{\mathrm{C}}_{\mathrm{2}} \:{and}\:\boldsymbol{\mathrm{C}}_{\mathrm{3}} \:{when} \\ $$ $$\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{1}} \boldsymbol{\mathrm{C}}_{\mathrm{2}} \:\:,\:\:\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} +\boldsymbol{\mathrm{r}}_{\mathrm{3}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{2}} \boldsymbol{\mathrm{C}}_{\mathrm{3}} \:\:{and} \\ $$ $$\boldsymbol{\mathrm{r}}_{\mathrm{3}} +\boldsymbol{\mathrm{r}}_{\mathrm{1}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{3}} \boldsymbol{\mathrm{C}}_{\mathrm{1}} . \\ $$ $${Note}\:{that}\:\boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{i}}} \boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{j}}} \:{is}\:{the}\:{distance}\:{between}\:{centres} \\ $$ $$\boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{i}}\:} {and}\:\:\boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{j}}} \:. \\ $$

Commented byYozzii last updated on 26/Dec/15

Commented byYozzii last updated on 26/Dec/15

Yes. I agree. Look at my image for a better idea on finding the full area.

$${Yes}.\:{I}\:{agree}.\:{Look}\:{at}\:{my}\:{image} \\ $$ $${for}\:{a}\:{better}\:{idea}\:{on}\:{finding}\:{the}\:{full} \\ $$ $${area}. \\ $$

Commented byRasheed Soomro last updated on 26/Dec/15

But the problem is to determine the central angles of these arcs!

$${But}\:{the}\:{problem}\:{is}\:{to}\:{determine}\:{the}\:{central}\:{angles} \\ $$ $${of}\:{these}\:{arcs}! \\ $$

Commented byYozzii 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.

$${It}\:{is}\:{possible}\:{that}\:{one}\:{circle}\:{completely} \\ $$ $${contains}\:{the}\:{other}\:{two}\:{overlapping} \\ $$ $${circles}.\:{The}\:{problem}\:{then}\:{requires} \\ $$ $${finding}\:{only}\:{the}\:{area}\:{in}\:{which}\:{the} \\ $$ $${two}\:{inner}\:{circles}\:{overlap}\:{since}\:{this} \\ $$ $${same}\:{area}\:{lies}\:{within}\:{the}\:{larger}\: \\ $$ $${circle}.\:{The}\:{conditions}\:{on}\:{r}_{{i}} \:{and}\:{C}_{{i}} {C}_{{j}} \\ $$ $${are}\:{still}\:{satisfied}.\: \\ $$ $$ \\ $$

Commented byYozzii 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 △ABC A_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 A_2 , A_3 and A_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_1 , where O_1 is the centre of this new circle, is A_2 , for example. The area of the region of overlap is then A_1 +A_2 +A_3 +A_4 .

$${I}\:{think}\:{I}\:{have}\:{a}\:{means}\:{of}\:{finding}\:{the}\:{required} \\ $$ $${area}\:{if}\:{no}\:{two}\:{circles}\:{fully}\:{contain}\:{each}\: \\ $$ $${other}.\:{It}\:{requires}\:{forming}\:{three}\:{new} \\ $$ $${circles}\:{and}\:{a}\:{triangle}.\:{A}\:{diagram}\: \\ $$ $${would}\:{truly}\:{help}\:{my}\:{explanation}.\: \\ $$ $$ \\ $$ $${Suppose}\:{no}\:{two}\:{circles}\:{fully}\:{contain} \\ $$ $${each}\:{other}\:{and}\:{that}\:{they}\:{appear}\:{to}\:{form} \\ $$ $${eight}\:{distinct}\:{regions}\:{in}\:{the}\:{plane}, \\ $$ $${with}\:{the}\:{common}\:{region}\:{of}\:{overlap}\:{having}\:{three} \\ $$ $${vertices}\:{A},{B},{C}.\:{Construct}\:{the}\:{triangle} \\ $$ $${A},{B},{C}.\:{Call}\:{the}\:{area}\:{of}\:\bigtriangleup{ABC}\:\boldsymbol{{A}}_{\mathrm{1}} . \\ $$ $${We}\:{have}\:{AB},{BC}\:{and}\:{CA}\:{as}\:{arcs}, \\ $$ $${because}\:{of}\:{the}\:{manner}\:{in}\:{which}\:{the} \\ $$ $${three}\:{cirvles}\:{overlap},\:{and} \\ $$ $$\:{we}\:{can}\:{form}\:{circles}\:{to}\:{adhere}\:{to} \\ $$ $${each}\:{of}\:{these}\:{arcs}.\:{Thus},\:{if}\:{this}\:{is}\:{done} \\ $$ $${we}\:{can}\:{find}\:{the}\:{respective}\:{areas}\:\boldsymbol{{A}}_{\mathrm{2}} , \\ $$ $$\boldsymbol{{A}}_{\mathrm{3}} \:{and}\:\boldsymbol{{A}}_{\mathrm{4}} ,\:{that}\:{lie}\:{between}\:{each}\:{side} \\ $$ $${of}\:{the}\:{triangle}\:{and}\:{the}\:{arc}\:{opposite}\: \\ $$ $${each}\:{side}.\:{For}\:{a}\:{given}\:{side}\:{with}\:{an}\:{arc}, \\ $$ $${the}\:{area}\:{difference}\:{between}\:{the}\:{sector} \\ $$ $${of}\:{the}\:{circle}\:{with}\:{arc}\:{AB},\:{for}\:{e}.{g},\: \\ $$ $${and}\:{the}\:{isosceles}\:{triangle}\:{ABO}_{\mathrm{1}} , \\ $$ $${where}\:{O}_{\mathrm{1}} \:{is}\:{the}\:{centre}\:{of}\:{this}\:{new}\: \\ $$ $${circle},\:{is}\:\boldsymbol{{A}}_{\mathrm{2}} ,\:{for}\:{example}.\:{The}\:{area}\:{of} \\ $$ $${the}\:{region}\:{of}\:{overlap}\:{is}\:{then}\: \\ $$ $$\boldsymbol{{A}}_{\mathrm{1}} +\boldsymbol{{A}}_{\mathrm{2}} +\boldsymbol{{A}}_{\mathrm{3}} +\boldsymbol{{A}}_{\mathrm{4}} .\: \\ $$ $$ \\ $$

Commented byRasheed Soomro last updated on 25/Dec/15

All possible cases should be considered separately

$${All}\:{possible}\:{cases}\:{should}\:{be}\:{considered} \\ $$ $${separately} \\ $$

Commented byYozzii 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! (Waste of energy...)

$${I}\:{just}\:{realized}\:{that}\:{you}\:{only}\:{need}\:{the} \\ $$ $${three}\:{initial}\:{circles}\:{to}\:{do}\:{the} \\ $$ $${calculation}.\:{The}\:{new}\:{circles}\:{I}\:{speak}\:{of} \\ $$ $${only}\:{need}\:{to}\:{be}\:{the}\:{initial}\:{circles}\:{we}\: \\ $$ $${started}\:{with}!\:\left({Waste}\:{of}\:{energy}...\right) \\ $$

Commented byRasheed Soomro last updated on 25/Dec/15

Three coplaner circles can form atmost seven distinct regions. I think.

$$\mathcal{T}{hree}\:{coplaner}\:{circles}\:{can}\:{form}\:\:{atmost} \\ $$ $${seven}\:{distinct}\:{regions}.\:{I}\:{think}. \\ $$

Commented byYozzii last updated on 25/Dec/15

The region outside all of them make it 8.

$${The}\:{region}\:{outside}\:{all}\:{of}\:{them}\:{make}\:{it}\:\mathrm{8}. \\ $$

Commented byRasheed Soomro last updated on 26/Dec/15

You should start with 7−region case , I think. I also think that the area of overlapping region is closely/direcly related to the quantities: r_1 +r_2 −C_1 C_2 , r_2 +r_3 −C_2 C_3 , r_3 +r_1 −C_3 C_1 More the circles are closed greater the area of overlapping region.

$${You}\:\:{should}\:{start}\:{with}\:\mathrm{7}−{region}\: \\ $$ $${case}\:,\:{I}\:{think}. \\ $$ $${I}\:{also}\:\:{think}\:{that}\:{the}\:{area}\:{of}\:{overlapping}\:{region}\:{is}\:{closely}/{direcly} \\ $$ $${related}\:{to}\:{the}\:{quantities}:\: \\ $$ $$\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} −\boldsymbol{\mathrm{C}}_{\mathrm{1}} \boldsymbol{\mathrm{C}}_{\mathrm{2}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} +\boldsymbol{\mathrm{r}}_{\mathrm{3}} −\boldsymbol{\mathrm{C}}_{\mathrm{2}} \boldsymbol{\mathrm{C}}_{\mathrm{3}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{3}} +\boldsymbol{\mathrm{r}}_{\mathrm{1}} −\boldsymbol{\mathrm{C}}_{\mathrm{3}} \boldsymbol{\mathrm{C}}_{\mathrm{1}} \\ $$ $${More}\:{the}\:{circles}\:{are}\:{closed}\:{greater}\:{the}\:{area}\: \\ $$ $${of}\:{overlapping}\:{region}. \\ $$

Commented byYozzii 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 △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.

$${A},{C}\:\&{E}\:{are}\:{circle}\:{centres}.\:{To}\:{find} \\ $$ $${the}\:{area}\:{between}\:{the}\:{arc}\:{HI}\:{and}\:{the} \\ $$ $${chord}\:{HI},\:{find}\:{the}\:{area}\:{of}\:\bigtriangleup{CHI} \\ $$ $${and}\:{then}\:{the}\:{area}\:{of}\:{the}\:{sector} \\ $$ $${CHI}.\:{The}\:{positive}\:{difference}\:{between} \\ $$ $${the}\:{two}\:{areas}\:{gives}\:{what}\:{we}'{re}\: \\ $$ $${searching}\:{for}.\:{This}\:{is}\:{similarly} \\ $$ $${done}\:{for}\:{sector}\:{EGH}\:{and}\:{sector}\:{AGI}. \\ $$ $$ \\ $$

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