Circumscribed Circle Calculator

Background.

Three points that do not sit on one straight line always lie on exactly one circle. Finding it is a practical job as often as a textbook one: locating the centre of a bolt pattern from three measured holes, or recovering the radius of a curved wall from three surveyed points.

Enter the points as x and y coordinates and you get the circle's radius, its centre in the same coordinate system, the circle's area and circumference, the triangle's own area, and whether the centre lies inside, on or outside that triangle. If you only know the three side lengths, the shape is fixed but its position is not, so the triangle is placed with corner A at the origin and side c along the positive x-axis. Radius and areas are unaffected by that choice; the centre coordinates are measured in it.

Three points on a line lie on no circle, two identical points lie on infinitely many, and side lengths where one equals or exceeds the other two combined never close into a triangle, so none of these give a result.

What is circumscribed circle calculator?

The circumscribed circle, or circumcircle, of a triangle is the circle passing through all three corners. Its centre, the circumcentre, is the point equally distant from the three vertices, and that common distance is the circumradius R. Euclid constructs it in Elements Book IV, Proposition 5: bisect two sides, raise perpendiculars at their midpoints, and the perpendiculars meet at the centre. It works because every point on a segment's perpendicular bisector is the same distance from both ends of that segment.

How to use this calculator.

1. 01Under "What do you have?" pick Three points for coordinates, or Three side lengths if you only measured the sides.
2. 02For points, enter x and y for A, B and C in one consistent unit; negatives and decimals are fine.
3. 03For sides, enter a, b and c, each named after the corner opposite it. Each must be shorter than the other two combined.
4. 04Read the centre as where the compass point goes and the radius as how far to open it.

The formula.

With coordinates, the centre comes from solving the two perpendicular-bisector equations at once. Let d = 2[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)] and sᵢ = xᵢ² + yᵢ². Then the centre is ux = [s₁(y₂ − y₃) + s₂(y₃ − y₁) + s₃(y₁ − y₂)] ÷ d and uy = [s₁(x₃ − x₂) + s₂(x₁ − x₃) + s₃(x₂ − x₁)] ÷ d, and R is the distance from (ux, uy) to any vertex. The quantity d is four times the triangle's signed area, so the triangle's area is |d| ÷ 4, and d = 0 is exactly the collinear case where no circle exists.

With side lengths, the triangle is first laid out as A = (0, 0), B = (c, 0) and C = (b cos A, b sin A), with cos A = (b² + c² − a²) ÷ 2bc from the law of cosines, and the same coordinate solution follows. The radius matches MathWorld's side-only formula R = abc ÷ √((a + b + c)(b + c − a)(c + a − b)(a + b − c)), which is abc ÷ 4K with the area K written out by Heron's formula.

The location line compares squared sides: with h the longest side and p, q the others, p² + q² − h² is positive for an acute triangle (centre inside), zero for a right triangle (centre on the hypotenuse's midpoint) and negative for an obtuse one (centre outside).

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — ‘Circumradius’ (live revision retrieved 2026-07-29). Equation (1), R = abc/√((a+b+c)(b+c−a)(c+a−b)(a+b−c)), and Equation (2), the semiperimeter form. Recorded honestly: the widely quoted R = abc/(4K) does not appear on that page in that arrangement — Equation (1) is the same identity with Heron's 4K written out — so this page cites what is actually printed. Equation (1) agrees with this calculator's coordinate-derived radius. Open access.
2. [2]Euclid, Elements, Book IV, Proposition 5 — ‘To circumscribe a circle about a given triangle’, constructed by bisecting two sides and intersecting the perpendiculars raised at their midpoints. The accompanying guide names the circumcircle, circumcentre and circumradius, and states that the centre falls inside acute triangles, on the hypotenuse of right triangles, and outside obtuse triangles — the source of this page's location output. David E. Joyce web edition of the Heath translation, Clark University; confirmed on retrieval 2026-07-29.
3. [3]Wolfram Research, MathWorld — ‘Heron's Formula’ (live revision retrieved 2026-07-29). Equation (2), Δ = √(s(s − a)(s − b)(s − c)). Used to cross-check the triangle area reported in side-lengths mode against the determinant used in coordinate mode, and to show that Equation (1) above is the familiar abc/4K. Open access.
4. [4]Wolfram Research, MathWorld — ‘Polygon Area’ (live revision retrieved 2026-07-29). Equation (4), A = ½ Σ (xᵢyᵢ₊₁ − xᵢ₊₁yᵢ). Source for the triangle-area determinant, which is also the exact collinearity test this page uses to reject three points that lie on one line. Open access.