## Circle-Circle Intersection

Let two Circles of Radii and and centered at and intersect in a Lens-shaped region. The equations of the two circles are

 (1) (2)

Combining (1) and (2) gives
 (3)

Multiplying through and rearranging gives
 (4)

Solving for results in
 (5)

The line connecting the cusps of the Lens therefore has half-length given by plugging back in to obtain
 (6)

giving a length of

 (7)

This same formulation applies directly to the Sphere-Sphere Intersection problem.

To find the Area of the asymmetric Lens'' in which the Circles intersect, simply use the formula for the circular Segment of radius and triangular height

 (8)

twice, one for each half of the Lens.'' Noting that the heights of the two segment triangles are
 (9) (10)

The result is

 (11)

The limiting cases of this expression can be checked to give 0 when and
 (12) (13)

when , as expected. In order for half the area of two Unit Disks () to overlap, set in the above equation
 (14)

and solve numerically, yielding .

© 1996-9 Eric W. Weisstein
1999-05-26