Enter circle coordinates and radii
Use any consistent unit system. The calculator compares center spacing, circumference clearance, containment, overlap, and lens area from two circles.
Example data table
These sample cases show how the same formulas behave across separate, tangent, intersecting, and contained circle layouts.
| Case | x₁ | y₁ | r₁ | x₂ | y₂ | r₂ | Center distance | Surface distance | Relationship |
|---|---|---|---|---|---|---|---|---|---|
| Separated | 0 | 0 | 3 | 12 | 0 | 4 | 12 | 5 | Separate circles |
| External tangent | 0 | 0 | 5 | 9 | 0 | 4 | 9 | 0 | Externally tangent circles |
| Intersecting | 1 | 2 | 6 | 8 | 4 | 5 | 7.2801 | 0 | Intersecting circles |
| Contained | 0 | 0 | 9 | 2 | 1 | 3 | 2.2361 | 3.7639 | One circle lies inside the other |
Formula used
The calculator evaluates spacing from the center coordinates first, then derives the appropriate surface distance based on whether the circles are apart, intersecting, tangent, or nested.
Core equations
Center distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Signed boundary separation: s = d − (r₁ + r₂)
External gap: max(0, d − r₁ − r₂)
Containment clearance: max(0, |r₁ − r₂| − d)
Minimum surface distance: 0 for tangent or intersecting cases; otherwise the needed positive clearance.
Geometry interpretation
If d > r₁ + r₂, the circles are separate.
If d = r₁ + r₂, they are externally tangent.
If |r₁ − r₂| < d < r₁ + r₂, they intersect.
If d = |r₁ − r₂|, they are internally tangent.
If d < |r₁ − r₂|, one circle lies fully inside the other.
How to use this calculator
- Enter the x and y coordinates for Circle 1.
- Enter Circle 1 radius using the same unit scale.
- Enter the x and y coordinates for Circle 2.
- Enter Circle 2 radius and choose decimal precision.
- Add a unit label such as cm, m, or px.
- Press Calculate Distance to show results above the form.
- Review the relationship, gap, overlap, and angle outputs.
- Use the CSV or PDF button to save the summary.
Frequently asked questions
1. What distance does this calculator report?
It reports the center distance and the minimum distance between circle boundaries. When circles intersect or touch, the minimum surface distance becomes zero.
2. What does a negative signed boundary separation mean?
A negative signed value means the circles overlap. Its magnitude shows how much combined radii exceed the center spacing along the line between centers.
3. Why can contained circles have zero overlap depth?
Overlap depth here measures penetration based on the sum of radii. Containment clearance separately measures how far the inner circle remains from the outer circumference.
4. Can I use negative coordinates?
Yes. Negative x and y values work normally because the formula depends on coordinate differences, not on whether coordinates are positive or negative.
5. Does the unit label affect calculations?
No. The unit label only describes the output. All calculations depend on the numeric inputs, so keep coordinates and radii in the same unit system.
6. What happens with concentric circles?
The calculator classifies them as coincident or contained concentric circles. In those cases, nearest perimeter points are not unique because every radial direction is equivalent.
7. When is the intersection area nonzero?
The intersection area becomes positive when circles overlap or one lies completely inside the other. It becomes zero for separated or externally tangent circles.
8. Why include the centerline angle?
The angle helps with plotting, CAD layouts, graphics, and directional geometry tasks. It shows the orientation from Circle 1 center toward Circle 2 center.