Free · Mathematical relations and geometry

Circle Equation from Diameter Endpoints

Find a circle center, radius and equation coefficients from the two ends of its diameter.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

Diameter endpoints (−1, −3) and (5, 5) have midpoint (2, 1), diameter 10 and radius 5. The equation is (x−2)²+(y−1)²=25.

center=((x₁+x₂)/2,(y₁+y₂)/2); r=√[(x₂−x₁)²+(y₂−y₁)²]/2

Worked example

Enter these known values and leave the other values blank.

x1
-1
y1
-3
x2
5
y2
5
A
2
B
1
C
25
A
2
B
1
r
5
D
-4
E
-2
F
-20
x
2
y
1
Radius
5
Diameter
10
Area
78.5
Circumference
31.4

Assumptions and limitations

  • The supplied points are opposite diameter endpoints, not arbitrary points on a circle.
  • Distinct endpoints determine a nondegenerate circle. Coincident endpoints yield zero radius and are outside the positive-radius model.
  • Coordinates and radius use one length scale. Area uses its square; no unit selectors are provided.
  • Coordinate inverses can have two geometric solutions. The positive square-root branch selects one solution, while midpoint relationships constrain linked coordinates.

Common questions

Can two arbitrary points on a circle determine its equation?

No. They determine this equation only when they are known opposite ends of a diameter. Many circles pass through two otherwise arbitrary points.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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