Enter Circle Data
Example Data Table
| Center 1 (x0,y0) | r0 | Center 2 (x1,y1) | r1 | Expected Type | Intersection Points (approx.) |
|---|---|---|---|---|---|
| (0, 0) | 5 | (6, 0) | 4 | Two points | (3.75, 3.307), (3.75, -3.307) |
| (0, 0) | 5 | (10, 0) | 5 | Tangent | (5, 0) |
| (0, 0) | 3 | (10, 0) | 2 | Separate | None |
Formula Used
d = √((x1−x0)² + (y1−y0)²) determines circle separation.
If d > r0 + r1, circles are separate. If d < |r0 − r1|, one contains the other.
For tangent or intersecting circles, compute:
- a = (r0² − r1² + d²) / (2d)
- h = √(r0² − a²)
- M = (x0 + a·dx/d , y0 + a·dy/d)
Intersection points:
- P1 = (Mx − dy·h/d , My + dx·h/d)
- P2 = (Mx + dy·h/d , My − dx·h/d)
Overlap area uses circular segment geometry and inverse cosine, producing the lens region when circles overlap.
How to Use This Calculator
- Enter the first circle center coordinates and its radius.
- Enter the second circle center coordinates and its radius.
- Click Calculate to view results above the form.
- Review the classification, distance, and any intersection points.
- Use the plot to visually confirm the geometry.
- Download a CSV or PDF report using the buttons.
Coordinate inputs and scaling
Circle intersections require consistent units across inputs. If centers are in meters, radii must be meters too. Very large coordinates can hide small separations, so translate the system near the origin when possible. A quick sanity check is verifying d, r0+r1, and |r0−r1| are on similar scales before solving. For CAD imports, keep coordinates in double precision and avoid rounding until final reporting.
Case classification with distance d
The calculator evaluates d=√((x1−x0)²+(y1−y0)²). If d>r0+r1, circles are separate. If d<|r0−r1|, one contains the other. If d≈0 and r0≈r1, circles coincide and have infinitely many intersection points. These rules also avoid division by 2d in degenerate cases. Near tangency, small floating errors appear; the tool clamps slightly negative h² values to zero.
Intersection point computation
For tangent or intersecting cases, the base offset a=(r0²−r1²+d²)/(2d) locates midpoint M along the center line. The height h=√(r0²−a²) sets a perpendicular displacement. Two-point solutions use ±(−dy·h/d, dx·h/d) added to M, producing P1 and P2 with symmetric geometry and identical radii checks. After solving, distances to each center are recomputed to confirm points satisfy both radii.
Chord length and diagnostic outputs
When two points exist, the chord length equals 2h. This diagnostic is valuable in QA: as overlap increases from near tangency, chord grows, reaches a maximum, and then shrinks as centers approach. Reporting a and h lets you audit intermediate steps, reproduce calculations in spreadsheets, and understand rounding impacts. Chord trends help flag outliers in repeated surveys and reveal unit mixups before final decisions.
Overlap area for planning tasks
Many workflows need the lens area, such as coverage overlap, tolerance stacking, or Venn-style comparisons. The calculator applies circular segment formulas using inverse cosine: each sector area minus its triangle area, summed across both circles. External tangency yields zero overlap. If one circle contains the other, overlap becomes the smaller circle’s area. Estimate shared coverage percentage by dividing overlap area by circle area, then comparing both ratios.
Visualization, exports, and verification
The plot shows both circles, centers, and any intersection points, so sign errors and swapped coordinates stand out immediately. CSV export supports downstream analysis, while PDF provides a shareable report that includes key metrics and a plot snapshot. For verification, test (0,0,5) and (10,0,5), which should return a single tangent point at (5,0). For publication, use the plot toolbar to export PNG images, then attach them to reports. Use the classification text to document assumptions when sharing results with teammates and reviewers later.
FAQs
What does “tangent circles” mean here?
Tangent circles touch at exactly one point. The calculator detects this when the computed height h becomes effectively zero, so only one intersection point is returned.
Why do I see “coincident circles”?
This happens when both centers match and both radii match. In that configuration, every point on the circle is an intersection, so unique coordinates cannot be listed.
Can the calculator handle negative coordinates?
Yes. Coordinates may be positive or negative because they represent a plane location. Radii, however, must be non‑negative; negative radii are rejected as invalid inputs.
Why are there no intersection points but overlap area shows a value?
If one circle fully contains the other, boundaries do not intersect, yet the region overlaps. In that case the overlap equals the smaller circle’s area, which the tool reports.
How accurate are the intersection coordinates?
Results use double‑precision arithmetic and rounding for display. Near tangency, tiny floating errors can occur, so the calculator clamps small negative h² values to zero.
What is the best way to verify my inputs?
Use the plot and the distance d. If d is much larger than r0+r1, circles should be separate. For a quick test, try a symmetric setup where you can predict the tangent or intersection points.