How to use the circle overlap area calculator
- Enter the radius of each circle, in any one unit.
- Enter the distance between the centers in the same unit.
- Read the overlap (lens) area and the union area.
- Check the shares of each circle that the overlap covers, and the shared chord length when the circles cross.
Formula
Lens = r1^2 acos((d^2 + r1^2 - r2^2) / (2 d r1)) + r2^2 acos((d^2 + r2^2 - r1^2) / (2 d r2)) - 0.5 sqrt((-d + r1 + r2)(d + r1 - r2)(d - r1 + r2)(d + r1 + r2))
Union = pi r1^2 + pi r2^2 - lens
No overlap when d is at least r1 + r2; full overlap when d is at most |r1 - r2|
acos is the inverse cosine in radians. The last term is the area of the kite formed by the two centers and the two crossing points.
Equal circles and a small circle inside
Two circles of radius 1 with centers 1 apart overlap in 2 pi / 3 - sqrt(3) / 2 = 1.2284 square units. Their union is 2 pi - 1.2284 = 5.0548. The overlap covers 39.1 percent of each circle.
A circle of radius 3 and one of radius 2 with centers 4 apart overlap in 1.9898 square units, with a union of 38.8509. If the smaller circle of radius 2 sits 2 from the center of a circle of radius 5, it lies entirely inside, and the overlap is its whole area, 12.5664.
Three cases
If the distance between centers is at least the sum of the radii, the circles do not meet and the overlap is zero. If it is at most the difference of the radii, the smaller circle is inside the larger and the overlap is its whole area. In between, the circles cross at two points and the overlap is a lens, which is the sum of two circular segments.
To find the crossing points themselves use the circle tangent and intersection calculator. For a single circle see the circle calculator.
Using the percentages
The share of the union is the ratio of the lens to the area covered by either circle, which is the overlap measure used to compare two shapes. The share of each circle shows how much of that circle the other one covers, which is handy for coverage questions such as sprinkler or signal ranges.
Assumptions and limits
- The circles are flat and perfect. Radii and distance use the same unit and the answers are in that unit squared.
- Zero distance between centers is allowed and gives full overlap of the smaller circle.
- Areas use pi to full double precision and are shown to four decimals.
- The shared chord is the straight line between the two crossing points and is shown only when the circles cross.
Frequently asked questions
How do I find the area where two circles overlap?
Use the lens formula with both radii and the distance between centers. It adds the two circular segments on either side of the shared chord, which is the formula this page evaluates.
What if one circle is completely inside the other?
Then the overlap is the whole smaller circle, pi times its radius squared. The page detects this when the distance is at most the difference of the radii.
Why is the overlap zero?
The circles are too far apart to touch, which happens when the distance between centers is at least the sum of the radii. Touching at a single point also gives zero area.
Can I use diameters instead of radii?
Halve each diameter first. The formula needs radii, and the distance between centers must be in the same unit as the radii.