Two Circle Boundary Intersection Calculator
Find boundary intersection coordinates of two circles and identify separate, contained, tangent, concentric and coincident cases.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Compare center distance with the radius sum and difference. For crossing circles, locate the common chord midpoint along the center line and offset perpendicularly.
a=(d²+rA²−rB²)/(2d); h=√(rA²−a²); intersections=chord midpoint±perpendicular h
Worked example
Two circles of radius 5 centered at (0,0) and (6,0) intersect at (3,4) and (3,-4), with common chord length 8.
Assumptions and input details
Cartesian coordinates and radii share one length unit. Positive radii 1e−9–1e9, center magnitudes up to 1e9. This calculates circle boundaries, not disk overlap area. Coincident circles have infinitely many boundary points; unequal concentric circles have none. Floating-point comparisons near tangency can be sensitive to rounded inputs, so near-tangent cases should be interpreted at input precision.
References: Wolfram Research: circle-circle intersections
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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