Area Of Multiple Overlapping Circles
Circle-Circumvolve Intersection
2 circles may intersect in ii imaginary points, a unmarried degenerate point, or ii distinct points.
The intersections of ii circles determine a line known as the radical line. If three circles mutually intersect in a single point, their point of intersection is the intersection of their pairwise radical lines, known as the radical center.
Allow two circles of radii 
and 
and centered at 
and 
intersect in a region shaped like an asymmetric lens. The equations of the two circles are
Combining (one) and (ii) gives
 | (iii) |
Multiplying through and rearranging gives
 | (iv) |
Solving for 
results in
 | (v) |
The chord connecting the cusps of the lens therefore has half-length 
given by plugging 
back in to obtain
Solving for 
and plugging back in to give the unabridged chord length 
then gives
This same formulation applies directly to the sphere-sphere intersection problem.
To find the area of the disproportionate "lens" in which the circles intersect, simply utilise the formula for the circular segment of radius 
and triangular superlative 
 | (10) |
twice, one for each half of the "lens." Noting that the heights of the ii segment triangles are
The upshot is
The limiting cases of this expression can be checked to give 0 when 
and
when 
, as expected.
In lodge for one-half the surface area of two unit disks (
) to overlap, set 
in the higher up equation
 | (17) |
and solve numerically, yielding 
(OEIS A133741).
If three symmetrically placed equal circles intersect in a single point, equally illustrated above, the total area of the three lens-shaped regions formed by the pairwise intersection of circles is given past
 | (xviii) |
Similarly, the full area of the four lens-shaped regions formed by the pairwise intersection of circles is given by
 | (19) |
See also
Borromean Rings, Brocard Triangles, Circle-Ellipse Intersection, Circumvolve-Line Intersection, Round Segment, Circular Triangle, Double Bubble, Caprine animal Problem, Johnson'southward Theorem, Lens, Lune, Mohammed Sign, Moss's Egg, Radical Heart, Radical Line, Reuleaux Triangle, Sphere-Sphere Intersection, Steiner Construction, Triangle Arcs, Triquetra, Venn Diagram, Vesica Piscis
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References
Sloane, N. J. A. Sequence A133741 in "The On-Line Encyclopedia of Integer Sequences."
Cite this as:
Weisstein, Eric W. "Circle-Circle Intersection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circle-CircleIntersection.html
Subject classifications
Area Of Multiple Overlapping Circles,
Source: https://mathworld.wolfram.com/Circle-CircleIntersection.html
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