How this calculator works
A diameter passes through the centre, so its midpoint fixes the centre and half its length fixes the radius. Substitution gives standard form; expanding the two squares gives general coefficients. Reverse mode reflects the known endpoint across the centre. Coordinates use the same scale on perpendicular axes. For other input combinations, use the related circle equation calculator.
Formula
h = (x₁+x₂)/2; k = (y₁+y₂)/2; r = hypot(x₂-x₁,y₂-y₁)/2. Standard: (x-h)²+(y-k)²=r². General: x²+y²+Dx+Ey+F=0, D=-2h, E=-2k, F=h²+k²-r². Opposite endpoint: (2h-x₁,2k-y₁).
Worked example
Endpoints (0,0) and (6,8) give midpoint (3,4) and radius 5. The circle is (x-3)²+(y-4)²=25, or x²+y²-6x-8y=0.
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Frequently asked questions
Can I enter any two points on a circle?
They must be opposite ends of a diameter. An arbitrary chord does not determine a unique circle.
Do the endpoint names matter?
Swapping them leaves the centre, radius and equation unchanged.
Why are coincident endpoints rejected?
They give zero radius, a single point rather than the positive-radius circle this tool solves.