What is a circumscribed circle?
A circumscribed circle, also called a circumcircle, passes through all three vertices of a triangle. Its center is the circumcenter, where the perpendicular bisectors of the triangle’s sides meet. Every triangle with three non-collinear vertices has exactly one circumcircle.
How to use this calculator
- Select Three side lengths or Vertex coordinates.
- Enter three positive sides that form a triangle, or the x and y coordinates of three distinct, non-collinear points.
- Click Calculate circumcircle to see the radius, diameter, area, circumference and center.
Side lengths determine the radius but do not specify the triangle’s position. In side mode, this calculator places A at (0, 0), B at (c, 0) and C above the x-axis. The reported center uses that placement. Use coordinate mode for the center of a triangle in your own coordinate system.
Circumscribed circle formula
For side lengths a, b and c, first find the semiperimeter s and triangle area Δ using Heron’s formula:
- Diameter: d = 2R
- Circle area: Acircle = πR²
- Circumference: Ccircle = 2πR
The side lengths must satisfy a + b > c, a + c > b and b + c > a. Equality produces a flat triangle with zero area and no finite circumcircle.
Circumcenter from coordinates
For vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃), translate A to the origin. Define u = x₂ − x₁, v = y₂ − y₁, p = x₃ − x₁ and q = y₃ − y₁. Let K = 2(uq − vp).
If K = 0, the points are collinear or repeated and there is no unique circumcircle. With center (Oₓ, Oᵧ), the circle equation is (x − Oₓ)² + (y − Oᵧ)² = R².
Worked example: 3–4–5 triangle
Take a triangle with sides a = 3 cm, b = 4 cm and c = 5 cm.
- Semiperimeter: s = (3 + 4 + 5) / 2 = 6 cm.
- Triangle area: Δ = √[6 × 3 × 2 × 1] = 6 cm².
- Circumradius: R = (3 × 4 × 5) / (4 × 6) = 2.5 cm.
- Diameter: 2 × 2.5 = 5 cm.
- Circle area: π × 2.5² ≈ 19.635 cm².
- Circumference: 2π × 2.5 ≈ 15.708 cm.
Using the side-mode placement, A = (0, 0), B = (5, 0) and C = (3.2, 2.4). The circumcenter is O = (2.5, 0), the midpoint of the hypotenuse.
Coordinate example
For A = (0, 0), B = (4, 0) and C = (0, 3), the circumcenter is (2, 1.5) and the radius is 2.5 units. The equation is (x − 2)² + (y − 1.5)² = 6.25. Both examples describe 3–4–5 right triangles in different placements.
Where is the circumcenter?
| Triangle type | Circumcenter location |
|---|---|
| Acute triangle | Inside the triangle |
| Right triangle | At the midpoint of the hypotenuse |
| Obtuse triangle | Outside the triangle |
| Equilateral triangle | At the common center; R = a / √3 |
Frequently asked questions
Is a circumcircle the same as an incircle?
No. A circumcircle passes through the triangle’s vertices. An incircle lies inside the triangle and touches all three sides.
Can the circumcenter be outside the triangle?
Yes. The circumcenter of an obtuse triangle lies outside the triangle.
What is the circumradius of a right triangle?
It is half the hypotenuse. For a hypotenuse of 10 cm, the circumradius is 5 cm.
Can three points on a straight line form a circumcircle?
No. Three distinct collinear points cannot lie on a finite circle. Enter three non-collinear points.
Can I find the center from only three side lengths?
You can find the center relative to an assumed placement of the triangle. To find its center in an existing coordinate system, enter the vertex coordinates.