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what is reuleaux triangle???
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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.
great.. i nvr studied this kind of triangle before. thanks for the explanation :D
its nice..
What is triangle?????
reuleaux triangle??? plz explain..in detail...
it so simble if we think in right method
good
ans. 6.3429
(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
what is meant by reuleaux triangle??
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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.
ughh i suck at this..... i guess its because im 11 its just my dream is hardvard
nice solution ivan martinez
what is reuleaux triangle???
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......
very good.
Very nice & clear explanation thank u
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
Really cool method dude :D
I did same too!
same here
i got the same solution
Very nice
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!
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
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 6 r 2 π . Area of triangle is 4 r 2 3 . After subtract are of triangle of circle slice we get 4 r 2 ( 2 π − 3 3 ) . After adding this 3 areas + area of triangle we get 2 9 ( π − 3 ) .... Probably it can simpler but this solution i found on quick....
see the angle coversd in one circle it is 60.
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)
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
that is simply using the equation 9/2 (pi-sq. root of 3)
3(9pi/6-9sqrt(3)/4)+9sqrt(3)/4
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.........
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.
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)
(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))
1/2(π-3^1/2)r^2 so, 1/2(π-3^1/2)3^2 = 9/2(π-3^1/2)
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)
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
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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)