Circumscribed Circle (Circumcircle) Calculator

About Circumscribed Circle (Circumcircle) Calculator

The Circumscribed Circle (Circumcircle) Calculator finds the circumscribed circle of any triangle. The circumscribed circle — also known as the circumcircle — is the unique circle that passes through all three vertices of a triangle. Enter three side lengths or three vertex coordinates to instantly compute the circumradius, circumcenter location, triangle area, interior angles, and more, with an interactive SVG diagram and step-by-step formulas.

Key Concepts of the Circumscribed Circle

Circumscribed Circle Formulas

For a triangle with sides a, b, c and semi-perimeter s = (a + b + c) / 2:

Circumcenter Location by Triangle Type

The position of the circumcenter depends on the type of triangle:

• Acute triangle: The circumcenter lies inside the triangle. All angles are less than 90°, so the perpendicular bisectors intersect within the triangle's interior.
• Right triangle: The circumcenter lies exactly at the midpoint of the hypotenuse. The circumradius equals half the hypotenuse length.
• Obtuse triangle: The circumcenter lies outside the triangle, on the opposite side from the obtuse angle. This is because the perpendicular bisectors diverge outward.

How to Find the Circumscribed Circle

1. Choose your input method: Select "Three Sides" if you know the side lengths a, b, c, or "Three Vertices" if you have the coordinates of each vertex.
2. Enter the values: Input the three side lengths or the (x, y) coordinates of vertices A, B, and C. Click a quick example to auto-fill sample values.
3. Click Calculate: Press the "Calculate Circumscribed Circle" button.
4. Review the results: See the circumradius R, circumcenter coordinates, circumcircle area and circumference, triangle area, angles, inradius, and the R/r ratio.
5. Explore the diagram: Toggle overlays for the circumcircle, perpendicular bisectors, radii, incircle, and labels to visualize the geometry.

Practical Applications

The circumscribed circle has important applications in many fields. In surveying and navigation, the circumcircle helps determine positions using triangulation. In computer graphics, Delaunay triangulation maximizes minimum angles by ensuring no vertex lies inside any triangle's circumcircle. In engineering, circumscribed circles define minimum enclosing boundaries for triangular components. The circumcircle is also fundamental in computational geometry algorithms for mesh generation and Voronoi diagrams.

Euler's Theorem and the Circumcircle

Euler's inequality states that for any triangle, the circumradius R is at least twice the inradius r: R ≥ 2r. Equality holds only for equilateral triangles. Additionally, Euler's formula relates the distance d between the circumcenter O and incenter I as \(d^2 = R(R - 2r)\). This elegant result connects the two most fundamental circles associated with a triangle and reveals deep properties of triangle geometry.

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