Circumscribed Circle Calculator
Calculate the circumscribed circle (circumcircle) of a triangle. Enter the three vertex coordinates to find the circumcenter and circumradius of the circle passing through all three vertices.
The Circumscribed Circle Explained
Every triangle can be inscribed in exactly one circle, called its circumscribed circle or circumcircle. This circle passes through all three vertices, and its center (the circumcenter) is equidistant from each vertex. The circumcenter is found at the intersection of the perpendicular bisectors of the triangle's sides, since any point on a perpendicular bisector is equidistant from the segment's endpoints.
The circumradius R can be calculated using the formula R = abc/(4K), where a, b, c are the side lengths and K is the triangle's area. An equivalent formula, R = a/(2sin A), relates the circumradius to any side and its opposite angle via the law of sines. These formulas connect the circumcircle to fundamental triangle properties and are used extensively in trigonometry and geometry.
Computing the Circumcenter
Given three vertex coordinates, the circumcenter is found by solving the system of perpendicular bisector equations. The perpendicular bisector of segment AB passes through the midpoint of AB and is perpendicular to AB. Writing equations for two perpendicular bisectors and solving their intersection gives the circumcenter coordinates.
The formula uses a determinant-based approach: D = 2(x1(y2-y3) + x2(y3-y1) + x3(y1-y2)). The circumcenter coordinates are then ratios of similar determinant expressions divided by D. When D equals zero, the three points are collinear and no circumcircle exists. This calculator implements the full formula and handles all valid triangle configurations.
Applications of the Circumcircle
The circumscribed circle has applications in computational geometry, particularly in Delaunay triangulation. The Delaunay condition requires that no point lies inside the circumcircle of any triangle in the mesh. This criterion produces triangulations that maximize the minimum angle, which is desirable for finite element analysis and terrain modeling.
In navigation, the circumcircle helps solve the problem of determining a position from three known landmarks. If you can measure the angles subtended by pairs of landmarks, the circumcircle relationship locates your position. In astronomy, circumscribed circles help calculate orbital parameters from three observed positions. The circumcircle also appears in the construction of the nine-point circle, which has a radius exactly half the circumradius.
Frequently Asked Questions
What is a circumscribed circle?
A circumscribed circle (circumcircle) is the unique circle that passes through all three vertices of a triangle. Its center is called the circumcenter and its radius is the circumradius.
Where is the circumcenter located?
The circumcenter is the intersection of the perpendicular bisectors of the triangle's sides. It is equidistant from all three vertices.
Is the circumcenter always inside the triangle?
No. It is inside for acute triangles, on the hypotenuse for right triangles, and outside for obtuse triangles.
What is the circumradius formula?
The circumradius R = abc/(4K) where a, b, c are the side lengths and K is the area. Alternatively, R = a/(2sin(A)) for any side a and its opposite angle A.
Does every triangle have a circumscribed circle?
Yes. Every non-degenerate triangle (three non-collinear points) has exactly one circumscribed circle. This is guaranteed by the fact that three non-collinear points determine a unique circle.