BIZOMAT MAKEOS · Geometry
Circle Through Three Points
What centre and radius pass through these three measured points?
Centre coordinates, radius and diameter of the circle through three points, for checking arcs, bores and bolt circles from coordinate measurements.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Point 1 X | mm | X coordinate of the first point. |
| Point 1 Y | mm | Y coordinate of the first point. |
| Point 2 X | mm | X of the second point. |
| Point 2 Y | mm | Y of the second point. |
| Point 3 X | mm | X of the third point. |
| Point 3 Y | mm | Y of the third point. Spread the points well around the arc for accuracy. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Point 1 X
- What it is
- The X coordinate of the first point measured on the arc, bore or bolt circle, in the same coordinate system as the other two points.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Spread the three points well around the circle; points close together make the centre very sensitive to error.
What this means: Point 1 Y
- What it is
- The Y coordinate of the first point measured on the arc, from the same origin as its X coordinate.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Negative values are allowed for points below the origin.
What this means: Point 2 X
- What it is
- The X coordinate of the second point measured on the arc.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same origin and axes for all three points.
What this means: Point 2 Y
- What it is
- The Y coordinate of the second point measured on the arc.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same origin and axes for all three points.
What this means: Point 3 X
- What it is
- The X coordinate of the third point measured on the arc.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The three points must not lie on a straight line.
What this means: Point 3 Y
- What it is
- The Y coordinate of the third point measured on the arc.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The three points must not lie on a straight line.
Method and formulas
- Solve the perpendicular bisectors: D = 2[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)].
- Centre X = [(x₁² + y₁²)(y₂ − y₃) + (x₂² + y₂²)(y₃ − y₁) + (x₃² + y₃²)(y₁ − y₂)] ÷ D, and similarly for Y.
- Radius = distance from the centre to any point.
- Example: the default points give a centre at (0, 0) and a radius of 50 mm.
Limitations
- Exactly three points define a circle but give no estimate of form error; use more points and a best fit for inspection.
- Points close together make the result very sensitive to measurement error.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find a bore or boss centre from three probe points
- Check an arc radius from coordinates
- Lay out a radius through known points
Who it is for
- Quality inspectors
- Toolmakers
- Metrology technicians
What you get
- Centre X and Y
- Radius and diameter
- Arc spread of the points
Interpreting the result
- Points spread over 120° or more give a reliable centre.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 40–48. Topic reference; no source tables or recommended settings are reproduced.
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