Two intersecting circles Watch

This discussion is closed.
Fegor
Badges: 0
#1
Report Thread starter 14 years ago
#1
This is a wierd one I set my self to keep my self awake over the summer but it is too hard for me.

Two identical circles overlap, so that each passes through the other's centre. What is the area in terms of r of the region which both circles ocupy.



a bit like this venn diagram but the circles are closer so that one goes through the other's centre and I want the area of region c on that diagram.

It looks like but I cant be sure that one of the points where the circles meet, and the two centres make an equalateral triangle. That would help.

thanks
0
sephonline
Badges: 0
#2
Report 14 years ago
#2
It's not hard. All you have to know is the formula for the area of a triangle, which is 1/2*a*b*sin c, as well as the area for sectors of a circle.
0
BCHL85
Badges: 12
Rep:
?
#3
Report 14 years ago
#3
(Original post by sephonline)
It's not hard. All you have to know is the formula for the area of a triangle, which is 1/2*a*b*sin c, as well as the area for sectors of a circle.
yeah, it's not hard.
Area = 4*integral ((-r) --> (-r/2)) of sqrt(r^2 - x^2).
0
jpowell
Badges: 15
Rep:
?
#4
Report 14 years ago
#4
Using calculus:

Circle 1: x^2 + y^2 = r^2

Circle 2: x^2 + (y-r)^2 = r^2

Substitute into each other and solve for x and y:

r^2 - y^2 = r^2 - (y-r)^2
y^2 = y^2 -2ry +r^2
y=r/2

therefore x = (sqrt(3)/2)*r and x = -(sqrt(3)/2)*r

Now, we need to only consider the upper half of circle 1 and the lower half of circle 2 so we get the formula's

y1=sqrt(r^2-x^2)
y2=r-sqrt(r^2-x^2)

Then using calculus to find the area under the graphs we get

A = Int{[From -(sqrt(3)/2)*r to (sqrt(3)/2)*r] of (y2 - y1)}

= Int{[From -(sqrt(3)/2)*r to (sqrt(3)/2)*r] of (r - 2*sqrt[r^2-x^2])}

= [rx -x*sqrt[r^2-x^2] +r^2*arcsin(x/r) from (-(sqrt(3)/2)*r to (sqrt(3)/2)*r)]

= [(sqrt(3)/2) - (2/3)*PI]*r^2

I'll try and tidy that up with some nice TeX later.
0
BCHL85
Badges: 12
Rep:
?
#5
Report 14 years ago
#5
Oh, it's quite long way . Just draw a diagram and see. Area C will be divided by 4 equal parts.
And consider a circle: x^2 + y^2 = r^2, then calculate one part by integral.. then time 4
0
jpowell
Badges: 15
Rep:
?
#6
Report 14 years ago
#6
Thats a good point.
0
Euler
Badges: 3
Rep:
?
#7
Report 14 years ago
#7
this kind of question is best done using polar coordinates and integration
0
RichE
Badges: 15
Rep:
?
#8
Report 14 years ago
#8
As sephonline said, it's not that hard and surely there's no need to resort to calculus. Not sure the person posing the question will have done integration.

The region comprises two segments, and the area of a segment is just the area of a sector minus a triangle.

The area of a sector subtending an angle theta (radians) is

1/2 r^2 theta

as it's theta/(2pi) of the whole circle's area. [If you've not met radians yet the formula is (angle in degrees/360) pi r^2]

And the angle at the centre is 2pi/3 as half the angle is subtended by a right-angled triangle with hypoteneuse r and adjacent r/2.
0
Euler
Badges: 3
Rep:
?
#9
Report 14 years ago
#9
Area of C = (r²)(2π/3 - √3/4)

it makes 4 segments and 2 equilateral triangles...

area of traingles =

(1/2)(r²)(sin 60)
(1/2)(r²)(√3/2)

area of both triangles is twice the above results :
(r²)(√3/2)

area of sectors is given by (1/2)r²(theta) where theta is in radians so area of each segment is area of sector minus area of triangle so total area of segments is:

2(r²π/3 - r²√3/2)

Adding the areas of two triangles and this results gives the required result
0
SsEe
Badges: 13
Rep:
?
#10
Report 14 years ago
#10
Area = [(2/3)π - √(3/4)]r²

Yep.
0
jpowell
Badges: 15
Rep:
?
#11
Report 14 years ago
#11
Nope, root(3)/2 is correct.
0
Euler
Badges: 3
Rep:
?
#12
Report 14 years ago
#12
(Original post by AntiMagicMan)
Nope, root(3)/2 is correct.
bah..yes it is
0
SsEe
Badges: 13
Rep:
?
#13
Report 14 years ago
#13
or root(3/4)
0
jpowell
Badges: 15
Rep:
?
#14
Report 14 years ago
#14
Of course, because they are the same thing. :rolleyes:
0
Fegor
Badges: 0
#15
Report Thread starter 14 years ago
#15
yes I know what integration is. I have the alevel.



thanks for that. I skim read your solutions so that I only got an idea of what you did, so I can do it myself and check it with yours later.

thanks
0
X
new posts
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

University open days

  • University of East Anglia
    All Departments Open 13:00-17:00. Find out more about our diverse range of subject areas and career progression in the Arts & Humanities, Social Sciences, Medicine & Health Sciences, and the Sciences. Postgraduate
    Wed, 30 Jan '19
  • Solent University
    Careers in maritime Undergraduate
    Sat, 2 Feb '19
  • Sheffield Hallam University
    City and Collegiate Campus Undergraduate
    Sun, 3 Feb '19

Brexit: Given the chance now, would you vote leave or remain?

Remain (553)
80.38%
Leave (135)
19.62%

Watched Threads

View All