Circle From 3 Points Calculator

How it is calculated

The circle through points P1, P2 and P3 is found with a determinant formula. First D = 2 × (x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)), which is four times the signed area of the triangle. If D is practically zero the points lie on one line, and the tool returns no result because no single circle exists. Otherwise, with sᵢ = xᵢ² + yᵢ², the centre is h = (s₁(y₂−y₃) + s₂(y₃−y₁) + s₃(y₁−y₂)) ÷ D and k = (s₁(x₃−x₂) + s₂(x₁−x₃) + s₃(x₂−x₁)) ÷ D. The radius is the distance from (h, k) to P1. Coordinates can be negative or decimal, and results are shown to three decimals in the same units you entered.

Common mistakes

• Entering three points on one straight line, such as (1,1), (2,2), (3,3). No unique circle passes through them, so the tool gives no result.
• Mixing up the order of x and y, for example typing x2 in the y2 box. The circle will still be computed, but for different points.
• Mixing units, such as one point in metres and another in millimetres. All six coordinates must use the same unit for the radius to make sense.
• Expecting an arc length or angle. The tool returns only the full circle's centre and radius, not the portion between your points.
• Using rounded measurements and trusting the third decimal. When the points are nearly collinear a tiny measurement error moves the centre a lot, so the radius becomes very sensitive.