Circle Through Three Points Calculator

Understanding the circle through three points — concept and principle

Any three points that do not lie on one straight line determine exactly one circle passing through all of them. This calculator finds that circle's centre and radius. Geometrically, the circle is the circumcircle of the triangle formed by the points and its centre is the circumcentre. It is used to reconstruct an arc from measured points, such as the radius of a curved road or a broken plate.

How it is calculated

Enter the points as "x y; x y; x y". With points A(ax, ay), B(bx, by), C(cx, cy):

• d = 2[ax(by − cy) + bx(cy − ay) + cx(ay − by)]. If d = 0, the points are collinear and no circle exists.
• Centre x: uₓ = [(ax² + ay²)(by − cy) + (bx² + by²)(cy − ay) + (cx² + cy²)(ay − by)] ÷ d.
• Centre y: u_y = [(ax² + ay²)(cx − bx) + (bx² + by²)(ax − cx) + (cx² + cy²)(bx − ax)] ÷ d.
• Radius: the distance from the centre to any of the points.

The principle

The centre must be equally far from all three points. Points equally far from A and B lie on the perpendicular bisector of AB, and points equally far from B and C lie on the perpendicular bisector of BC. Where the two bisectors cross, the point is equidistant from all three, so it is the centre. Setting the squared distances equal gives two linear equations in the centre's coordinates, and Cramer's rule solves them. The value d equals four times the triangle's signed area, so it is zero exactly when the points are collinear.

Worked example

1. Inputs: A = (0, 0), B = (4, 0), C = (0, 3).
2. d = 2[0 × (0 − 3) + 4 × (3 − 0) + 0 × (0 − 0)] = 24.
3. Squared lengths: A → 0, B → 16, C → 9. uₓ numerator: 0 × (0 − 3) + 16 × (3 − 0) + 9 × (0 − 0) = 48, so uₓ = 48 ÷ 24 = 2.
4. u_y numerator: 0 × (0 − 4) + 16 × (0 − 0) + 9 × (4 − 0) = 36, so u_y = 36 ÷ 24 = 1.5. Centre (2, 1.5).
5. Radius: √((0 − 2)² + (0 − 1.5)²) = √6.25 = 2.5.

How to read the result

Here the triangle has a right angle at the origin, so by Thales' theorem the centre is the midpoint of the hypotenuse BC and the radius is half its length, 5 ÷ 2 = 2.5. For an acute triangle the centre lies inside, for an obtuse triangle outside. If the points are close to a straight line, the radius becomes very large and sensitive to measurement errors.

Common mistakes

• Entering the points in the wrong format; use a space between x and y and a semicolon between points.
• Using three nearly collinear points from a gentle curve, which makes the result unstable.

Frequently asked questions

Why does a straight line of points give no circle?

Their perpendicular bisectors are parallel and never meet, so no point is equidistant from all three.

How do I find the radius of an arc from its chord and height?

With chord length L and arc height s, r = (L²/4 + s²) ÷ (2s).

How do I check whether four points lie on one circle?

Find the circle through three of them and check whether the fourth is at the same distance from the centre.

Sources: Euclid, Elements, Book IV, Proposition 5 (circumscribing a circle about a triangle) · D. Zwillinger (ed.), CRC Standard Mathematical Tables and Formulae, CRC Press