Layout and shop math
Circle center from 3 points calculator
Enter three X, Y points on a bore or bolt circle, as read off the DRO, to get the center, radius and diameter.
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
Center X (Ux)
in
Ux = (S1 x (y2 - y3) + S2 x (y3 - y1) + S3 x (y1 - y2)) / K = (13.25 x (2.6 - (-0.2)) + 6.85 x ((-0.2) - 1) + 0.05 x (1 - 2.6)) / 19.2 = 1.5000 in
K = 2 x (x1 x (y2 - y3) + x2 x (y3 - y1) + x3 x (y1 - y2)) = 2 x (3.5 x (2.6 - (-0.2)) + 0.3 x ((-0.2) - 1) + (-0.1) x (1 - 2.6)) = 19.2 sq in
Center Y (Uy)
in
Uy = (S1 x (x3 - x2) + S2 x (x1 - x3) + S3 x (x2 - x1)) / K = (13.25 x ((-0.1) - 0.3) + 6.85 x (3.5 - (-0.1)) + 0.05 x (0.3 - 3.5)) / 19.2 = 1.0000 in
S1 = 13.25 sq in; S2 = 6.85 sq in; S3 = 0.05 sq in
Diameter
in
r = sqrt((x1 - Ux)^2 + (y1 - Uy)^2) = sqrt((3.5 - 1.5)^2 + (1 - 1)^2) = 2.0000 in
Radius 2.0000 in. Diameter 4.0000 in (101.60 mm).
Source: Machinery's Handbook (circle through three points); standard analytic geometry
How it works
K = 2 x (x1 x (y2 - y3) + x2 x (y3 - y1) + x3 x (y1 - y2)); Sn = xn^2 + yn^2; Ux = (S1 x (y2 - y3) + S2 x (y3 - y1) + S3 x (y1 - y2)) / K; Uy = (S1 x (x3 - x2) + S2 x (x1 - x3) + S3 x (x2 - x1)) / K; r = sqrt((x1 - Ux)^2 + (y1 - Uy)^2)
| Symbol | Meaning | Unit |
|---|---|---|
| x1, y1 to x3, y3 | The three measured points | in or mm |
| K | Four times the signed area of the triangle through the three points; zero when the points are in a line | sq in or sq mm |
| S1, S2, S3 | x squared plus y squared for each point | sq in or sq mm |
| Ux, Uy | Circle center | in or mm |
| r | Radius, the distance from the center to any of the points | in or mm |
Worked example
Three holes of a bolt circle picked up with a coaxial indicator. DRO readings: (3.500, 1.000), (0.300, 2.600) and (-0.100, -0.200) in.
- K = 2 x (3.5 x (2.6 - (-0.2)) + 0.3 x (-0.2 - 1.0) + (-0.1) x (1.0 - 2.6)) = 2 x (9.8 - 0.36 + 0.16) = 19.2 sq in.
- S1 = 3.5^2 + 1.0^2 = 13.25 sq in; S2 = 0.3^2 + 2.6^2 = 6.85 sq in; S3 = (-0.1)^2 + (-0.2)^2 = 0.05 sq in.
- Ux = (13.25 x 2.8 + 6.85 x (-1.2) + 0.05 x (-1.6)) / 19.2 = (37.1 - 8.22 - 0.08) / 19.2 = 28.8 / 19.2 = 1.5000 in.
- Uy = (13.25 x (-0.4) + 6.85 x 3.6 + 0.05 x (-3.2)) / 19.2 = (-5.3 + 24.66 - 0.16) / 19.2 = 19.2 / 19.2 = 1.0000 in.
- r = sqrt((3.5 - 1.5)^2 + (1.0 - 1.0)^2) = sqrt(4) = 2.0000 in, so the diameter is 4.0000 in.
Result: center X 1.5000 in, Y 1.0000 in; bolt circle diameter 4.0000 in. Each point checks out at 2.0000 in from the center.
Shop notes
- Bolt circle: pick up the center of three holes with a coaxial indicator or a pin in the spindle, and enter those readings.
- Bore: touch the wall at three points with an edge finder or probe. The center comes out right; add the finder radius to the radius for the bore size.
- Spread the points around the circle. Three points close together magnify any reading error.
- Use the same DRO zero for all three points. Moving the zero between readings moves the answer.
FAQ
How do you find the center of a circle from 3 points?
Compute K = 2 x (x1 x (y2 - y3) + x2 x (y3 - y1) + x3 x (y1 - y2)), then Ux and Uy from the sums of squares divided by K. For points (3.500, 1.000), (0.300, 2.600) and (-0.100, -0.200) in, K = 19.2 sq in and the center is X 1.5000 in, Y 1.0000 in, radius 2.0000 in.
Why does the calculator say the points are in a line?
K comes out zero (or close to it) when the three points sit on one straight line. No circle passes through them. Pick points spread around the circle.
Does the edge finder or probe size matter?
Not for the center: three spindle positions taken with the same edge finder inside a bore sit on a circle with the same center as the bore. The radius comes out smaller by the edge finder radius, so add the finder radius to get the bore radius.
How far apart should the three points be?
Spread them around the circle, ideally about 120° apart. Points bunched close together turn a small reading error into a large center error.
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