Circle-Circle Intersection

A circle-circle intersection consists of the points common to two circles. Two distinct circles can have zero, one, or two real points of intersection. Coincident circles have their entire circumference in common. For nonconcentric circles with no real intersection, the equations have two nonreal conjugate solutions.

The radical line of two nonconcentric circles is the locus of points of equal circle power. If the circles intersect in two points, it is the line through them. If three circles with noncollinear centers have a common point, their pairwise radical lines meet at that point, the radical center.

Let two circles of radii and , centered at and , intersect in two distinct points, so . Their corresponding closed disks have an overlap 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 two points of intersection are the corners of the lens. Their connecting chord has half-length given by plugging back in to obtain

			(6)
			(7)

Solving for and plugging back in to give the entire chord length then gives

			(8)
			(9)

This same formulation applies directly to the sphere-sphere intersection problem.

The interior angle of the lens at either intersection and the angle between the corresponding radii are supplementary angles. The law of cosines therefore gives

(10)

for .

To find the area of the overlap of the two closed disks, use the formula for the circular segment of radius and signed offset , with

(11)

for each of the two circular segments. Their signed offsets are

			(12)
			(13)

The result is

			(14)
			(15)

For , the closed disks have overlap area 0. For , their overlap area is . The preceding formula applies in the partial-overlap case and extends by limits to tangency. Setting gives

			(16)
			(17)

for .

In order for the overlap of two unit disks () to have half the area of each disk, set in the above equation

(18)

and solve numerically, yielding (OEIS A133741). A representation in terms of the inverse function of the regularized beta function is

(19)

where is the regularized beta function and is its inverse function with respect to (Fang 2025, p. 5).

For three unit circles whose centers are the vertices of an equilateral triangle of circumradius 1, the common point is the circumcenter. The three circles form an equilateral configuration of Johnson circles. The three pairwise overlap regions of the corresponding unit disks have disjoint interiors, with total area

(20)

For four unit circles whose centers are the vertices of a square of circumradius 1, the four adjacent-pair overlap regions of the corresponding unit disks have disjoint interiors, with total area

(21)