Circumscribed Circle: Circumradius, Circumcenter, and Triangle Geometry
Calculate the circumscribed circle of any triangle with our free tool. Find the circumradius, circumcenter, and circle equation from side lengths or coordinates.
The circumscribed circle (circumcircle) of a triangle is the unique circle that passes through all three vertices of the triangle. Its center, called the circumcenter, is the point equidistant from all three vertices. Finding the circumcircle of a triangle connects several fundamental concepts in geometry: perpendicular bisectors, the Pythagorean theorem, the law of sines, and coordinate geometry. From GPS triangulation to architectural arch design, the circumscribed circle is a powerful geometric tool.
Key Takeaway
The circumradius of a triangle with sides a, b, c and area K is R = abc/(4K). The circumcenter is the intersection of the perpendicular bisectors of the triangle's sides. For a right triangle, the circumcenter is at the midpoint of the hypotenuse.
1. What Is a Circumscribed Circle?
A circumscribed circle is the smallest circle that completely contains a triangle, with all three vertices lying exactly on the circle. Every triangle has exactly one circumcircle (unlike some polygons that may not have one). The circumcenter — the center of this circle — is always equidistant from all three vertices.
The position of the circumcenter relative to the triangle depends on the triangle's angles: inside for acute triangles, on the hypotenuse midpoint for right triangles, and outside for obtuse triangles.
2. The Circumradius Formula
The circumradius R of a triangle with sides a, b, c and area K is:
Where K can be computed using Heron's formula: K = √(s(s−a)(s−b)(s−c)), with s = (a+b+c)/2.
This formula is intimately connected to the law of sines: a/sin A = b/sin B = c/sin C = 2R. Rearranging gives R = a/(2 sin A), providing an alternative way to compute the circumradius when angles are known.
3. Finding the Circumcenter
The circumcenter is found at the intersection of the perpendicular bisectors of the triangle's sides. Each perpendicular bisector is the locus of points equidistant from two vertices. The intersection of any two perpendicular bisectors determines the circumcenter (the third bisector automatically passes through the same point).
In coordinate geometry, the circumcenter (x₀, y₀) satisfies |OA|² = |OB|² = |OC|², where O is the circumcenter and A, B, C are the vertices. This yields two linear equations that can be solved for x₀ and y₀.
4. Special Cases
| Triangle Type | Circumcenter Location | Circumradius |
|---|---|---|
| Equilateral (side a) | At centroid (inside) | R = a/√3 |
| Right (legs a, b; hyp c) | Midpoint of hypotenuse | R = c/2 |
| Obtuse | Outside the triangle | R > longest side/2 |
5. Applications
The circumscribed circle is used in GPS triangulation (finding a position from three reference points), structural engineering (finding the center of circular arches), CNC machining (fitting circular toolpaths through three points), surveying (determining boundary arcs), and computer graphics (fitting circles to three points for rendering curves).
6. Frequently Asked Questions
Can the circumcenter be outside the triangle?
Yes. For obtuse triangles, the circumcenter lies outside the triangle, on the side opposite the obtuse angle. For right triangles, it lies exactly on the hypotenuse midpoint.
What is Euler's line?
Euler's line passes through the circumcenter, centroid, and orthocenter of a triangle. These three points are always collinear, with the centroid dividing the segment from circumcenter to orthocenter in ratio 1:2.
Using Our Calculator
Enter the three side lengths of any triangle to instantly find the circumradius, circumcenter, triangle area, and circle equation. The calculator validates that the sides form a valid triangle before computing.