Area of intersection between two circles

[Sarah L]

Hi,

I would very much like someone to help me solve the area of intersection between to intersecting circles (one with the radius r, and one with the radius 1). Tangents at the intersecting point form a 120 degree outer angle.

1. Homework Statement , 2 Relevent equations

Here is a sketch of the problem: http://i42.tinypic.com/m9254i.jpg
I want to calculate the area of the intersection between the two circles.

The Attempt at a Solution

I've tried to calculate the distance between the centres of the two circles and thought I could use that to somehow calculate the area of the intersection but I haven't managed to find any solution to the problem.


Thank you so much for your time and help.


Sarah

 

[haruspex]

Hi Sarah,
If you draw a line joining the two points of intersection, you will cut the area into two segments. Can you see how to calculate the area of one of these segments (as the difference of two simpler areas)?

 

[lurflurf]

Concentrate on the sectors of intersection (there is no intersection in the rest of the circles).

Area intersection=Area sector 1+Area sector 2-Area union of sectors
The union of sectors is the kite (quadrilateral with two pairs of adjacent equal sides) the two sectors form
The idea is the overlap is the double conted area so we can add the two areas and subtract the single counted part to find it.

usefull formula
area sector=(angle/angle full circle)pi r^2
area kite=rR sin (angle)
r R radii two circles
angle angle between unequal sides
length of chord 2r sin(angle/2)

 

[verty]

The first thing is to find angles MCO and MOC in terms of r.