3 Overlapping Circles

Geometry Level 1

5 ( π 3 ) 5 \left( \pi-\sqrt{3} \right) 4 ( π 3 ) 4 \left( \pi-\sqrt{3} \right) 11 2 ( π 3 ) \frac{11}{2} \left( \pi-\sqrt{3} \right) 9 2 ( π 3 ) \frac{9}{2} \left( \pi-\sqrt{3} \right)

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19 solutions

Ivan Martinez
Mar 2, 2014

The area of a reuleaux triangle is: 1/2(π-3^1/2)r^2 therefore: 1/2(π-3^1/2)3^2 = 9/2(π-3^1/2)

what is reuleaux triangle???

Archit Murkunde - 7 years, 3 months ago

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The shape of constant radius made by three overlapping circles. It's the figure that you find the area of in this problem.

Daniel Bortolussi - 7 years, 3 months ago

great.. i nvr studied this kind of triangle before. thanks for the explanation :D

Heng Joe Kit - 7 years, 3 months ago

its nice..

Vema Venki - 7 years, 3 months ago

What is triangle?????

Rafaqat Ali - 7 years, 3 months ago

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a shape with three angles

Thích Thảo - 7 years, 3 months ago

reuleaux triangle??? plz explain..in detail...

Gv Nikhil - 7 years, 3 months ago

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see en.wikipedia.org/wiki/Reuleaux_triangle

Aviral Rastogi - 7 years, 3 months ago

it so simble if we think in right method

Melvin Mosq - 7 years, 3 months ago

good

Melvin Mosq - 7 years, 3 months ago

ans. 6.3429

Dean Clidoro - 7 years, 3 months ago

(3Api*r^2)/360 ===3 sectors
then subtract 2 triangles inscribe in the shaded area

2(d1d1sinA/2) === area of 2 inscribed triangles in the shaded area

ans.6.3429

Dean Clidoro - 7 years, 3 months ago

what is meant by reuleaux triangle??

Dhanya Nedungadi - 7 years, 2 months ago

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A Reuleaux triangle is the simplest and best known Reuleaux polygon. It is a curve of constant width, meaning that the separation of two parallel lines tangent to the curve is independent of their orientation. Because all diameters are the same, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?" The term derives from Franz Reuleaux, a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another, although the concept was known before his time. With a compass, sweep an arc sufficient to enclose the desired figure. With radius unchanged, sweep a sufficient arc centred at a point on the first arc to intersect that arc. With the same radius and the centre at that intersection sweep a third arc to intersect the other arcs. The result is a curve of constant width. Equivalently, given an equilateral triangle T of side length s, take the boundary of the intersection of the disks with radius s centered at the vertices of T. By the Blaschke–Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width. This area is {1\over2}(\pi - \sqrt3)s^2, where s is the constant width. The existence of Reuleaux polygons shows that diameter measurements alone cannot verify that an object has a circular cross-section. The area of Reuleaux triangle is smaller than that of the disk of the same width (i.e. diameter); the area of such a disk is \pi s^2 \over 4.

Melvin Mosq - 7 years, 2 months ago

ughh i suck at this..... i guess its because im 11 its just my dream is hardvard

bob jerry - 7 years, 2 months ago

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Harvard* lol

Vasudev Chandna - 6 years, 2 months ago

nice solution ivan martinez

Muhammed Attaul Gani Chowdhury - 7 years, 1 month ago

what is reuleaux triangle???

Christian Cernechez - 7 years, 3 months ago
Ana Paula Mello
Mar 3, 2014

You can connect the center of each circle to make a triangle inside the shaded area. Since the sides of the triangle are the radius of the circles (measuring 3 each), you have an equilateral triangle, with all angles measuring 60 degrees.

You can also get an arc from each circle inside the same area. The angle of the arc is the same as the angle of the triangle, which is 60 degrees. By putting the arcs together, you get to the shaded area you want, with the following equation for the area:

A=3 (pi r^2*angle)/360

But there's a problem; the arc will have overlapping areas. Luckily, these areas are equivalent to the equilateral triangle from earlier, so all you have to do is subtract the area of the triangle twice to get rid of the overlapping, and you have the final equation to solve the problem:

A=[3 (pi r^2 angle)/360]-[2 (r^2*sqrt3)/4]

Plug in the numbers and you'll have 9/2 (pi-sqrt3)

very nice explanation bro......

Sanjeev Prasad - 7 years, 3 months ago

very good.

Jess Toni Bautista - 7 years, 2 months ago

Very nice & clear explanation thank u

Matheswari selvaraj - 7 years, 2 months ago
Aji Karunakaran
Mar 4, 2014

I bumped into the answer this way:-

We know that, Segment = Sector - Corresponding Triangle.

Area of the shaded region is = Area of ONE Sector + Area of TWO Segments
= Area of THREE Sectors - Area of TWO Triangles.

Hope you understand. Am not able to type-in the calculations here. See Ana Paula Mello's answer below.

ur method is good

Mian Fahad - 7 years, 3 months ago

Really cool method dude :D

Malhar Savale - 7 years, 3 months ago

I did same too!

Kou$htav Chakrabarty - 7 years, 3 months ago

same here

Ankit Tiwari - 7 years, 3 months ago

i got the same solution

David Lancaster - 7 years, 3 months ago

Very nice

Matheswari selvaraj - 7 years, 2 months ago
Jeeves Techie
Mar 8, 2014

I it quite simple if we see it geometrically. If you join the three point using straight lines, then you will obtain a equilateral triangle whose side will be equal to the radius of the individual.Let its area be delta which can be calculated using a simple formula of area of a equilateral traingle. If you take one point as centre and look upon the figure then other two points will look as if they are subtending the arc on the point making an angle of 60 degree. We can calculate the are of the arc using sinple arc formula. Let its area be arc. Now we can see that area of the remaining portion will be arc-delta. So the area of the shaded portion will be equal to 3*(arc-delta)+delta

I not the only one!

Julian Poon - 7 years, 2 months ago
Naka Ra
Mar 5, 2014

The area of a reuleaux triangle is: 1/2(π-3^1/2)r^2 therefore: 1/2(π-3^1/2)3^2 = 9/2(π-3^1/2)

WRONG

Jaan Khan - 7 years, 3 months ago

When you connect centers of circle you get equilateral triangle. Equilateral triangle have all three angles equal and it is 60 degrees. This angle represents sixth of area of each circle. So it is r 2 π 6 \frac{r^2\pi}{6} . Area of triangle is r 2 3 4 \frac{r^2\sqrt{3}}{4} . After subtract are of triangle of circle slice we get r 2 ( 2 π 3 3 ) 4 \frac{r^2(2\pi-3\sqrt{3})}{4} . After adding this 3 areas + area of triangle we get 9 2 ( π 3 ) \frac{9}{2}(\pi-\sqrt{3}) .... Probably it can simpler but this solution i found on quick....

Sai Sankalp
Mar 20, 2014

see the angle coversd in one circle it is 60.

David Lancaster
Mar 13, 2014

The needed figure shows 3 overlapping sectors.

To get the area of it simply get the area of 3 sectors minus the area of 2 equilateral triangle. Theta or the angle to be used is 60 ° = π/3 in radians

Use the formula below

Area of a sector = (1/2 r^2∅) or (1/2 r^2 π/3)

Multiplying this to 3 we get

1/2 r^2 π

The area of an equilateral triangle is

(r^2 √3)/4

Getting 2 of this will result to

(r^2 √3)/2

Area of the reuleaux triangle is

(r^2 π)/2 - (r^2 √3)/2

Simplify

r^2/2 (π-√3)

Substituting the value of the radius we get the answer

9^2/2 (π-√3)

Saurabh Gupta
Mar 13, 2014

If we join all the three centres ( dots in the figure ) we will get an equilateral triangle. use the formula of finding the area of the sector with Theta as 60 degrees. this value comes out to be 60/360 * pi * 9. Subtract the area of the equilateral triangle so that we find the other left out area of one part. Multiply this by 3 3 ( 60/360 * pi * 9 - root3/ 4 * 9) + root3/4 * 9 this will give you the asnwer

Job Triviño
Mar 13, 2014

that is simply using the equation 9/2 (pi-sq. root of 3)

3(9pi/6-9sqrt(3)/4)+9sqrt(3)/4

Subhrajyoti Sinha
Mar 12, 2014

3x{(60/360)x9pi - (3^(1/2)/4)x9}+(3^(1/2)/4)x9=9/2(pi-3^(1/2))................here (3^(1/2)/4)x9 is the area of the equilateral triangel, made by joining the points where the circles intersect.........

Melvin Mosq
Mar 12, 2014

first you must understand that there is an equilateral triangle with radius 3 inscribed in the shaded area.so you first find that and find the sectors area with angle 60.Then substract the area of triangle from area of sector then you will get a new area .multiply it by 3 and add the quantity with area of the triangle .hence the answer.

Manjunath Misal B
Mar 10, 2014

can use s=r theta & a=1/2 r^2*theta formula along wid area of equilateral triangle formula & compare.. ans:9/2(pi-root(3))

1/2(π-3^1/2)3^2 = 9/2(π-3^1/2)

Hitoshi Yamamoto
Mar 8, 2014

(π.r^2)/2-2.( ( r^2.3^(1/2) )/4) = (π.9)/2 - ((9.3^(1/2))/2 = 9/2(π - 3^(1/2))

Mohsin Iqbal
Mar 8, 2014

1/2(π-3^1/2)r^2 so, 1/2(π-3^1/2)3^2 = 9/2(π-3^1/2)

Aditya Gupta
Mar 8, 2014

The area of a reuleaux triangle is: 1/2(π-3^1/2)r^2 therefore: 1/2(π-3^1/2)3^2 = 9/2(π-3^1/2)

Saumitra Paira
Mar 7, 2014

see,if u join the intersection points mutually, u get a equilateral triangle of radius 3 cm, so first find out the area of this triangle=(sqr root(3)/4) (3 3) = 9 root(3)/4 now if u see the triangle carefully, you will see an arc of 60 degree in each triangle, so lets first find out the area subtended by this arc to the centre of the circle=pi 3 3 (60/360). note that u are considering one circle. subtract the area of the triangle from this area and multiply it by 3 to do it for all the three circles.so we get = 3 (9 pi/6-9*root(3)/4) . now add the area of the triangle to get the shaded area=9/2(pi-root(3)).
Thanks

As all circle intersect at the center of other, by joining three point an equilateral triangle will be formed whose area is (3^1/2) (3^2)/4. If we consider one circle at time considering point intersection, it will form a sector of 60 degree angle whose area is (60/360) pi (3^2). Now, area of shaded region = 3 area of sector - 2*area of equilateral triangle

Balram Suhane - 7 years, 3 months ago

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