Circle–Circle Intersection

Drag the circles. Classification by d vs r1+r2 / |r1-r2|, the a/h construction, and the lens area.

Classify: O(1) Intersect (a/h): O(1) Lens area: O(1)
120 120
Circle 1 Circle 2 Intersection Lens a / h

Tip: drag a circle's center dot to move it. Use the sliders to resize.

Classification & Geometry

d = |c1 - c2|
r1 + r2
|r1 - r2|
a = (d²+r1²-r2²)/(2d)
h = sqrt(r1²-a²)
# intersection points
lens (overlap) area
point P+
point P-

Complexity

OperationTimeNote
Center distance dO(1)1 hypot
ClassifyO(1)2 comparisons
Intersect (a/h)O(1)1 sqrt
Lens areaO(1)2 acos
Overlap test (no sqrt)O(1)d² ≤ (r1+r2)²

Operation Log