Circumscribed Circle Calculator

Find the circumscribed circle (circumcircle) of a triangle from 3 sides or 3 vertices. Calculates the circumradius R = abc / (4K).

Enter Triangle Data

Results

Circumradius (R)
3.57143
units
Triangle Area (K)14.6969
Semi-perimeter (s)9
Circumference of Circumcircle22.4399
Area of Circumcircle40.0713
The given sides do not form a valid triangle.

Step-by-Step Solution

R = abc / (4K)

What is a Circumscribed Circle?

A circumscribed circle (also called a circumcircle) of a triangle is the unique circle that passes through all three vertices of the triangle. The center of the circumscribed circle is called the circumcenter, and the radius is called the circumradius (R). Every triangle has exactly one circumscribed circle.

The circumcenter is equidistant from all three vertices of the triangle and lies at the intersection of the perpendicular bisectors of the triangle's sides.

Key Formulas

Circumradius from Sides

Using Heron's formula for area K and side lengths a, b, c.

R = abc / (4K)

Heron's Formula

Area from three side lengths using the semi-perimeter s.

K = sqrt(s(s-a)(s-b)(s-c))

Semi-perimeter

Half the sum of all three sides.

s = (a + b + c) / 2

Extended Law of Sines

Relates circumradius to sides and opposite angles.

a/sin(A) = 2R

The Circumcenter

The circumcenter of a triangle has special properties depending on the type of triangle:

  • Acute triangle: The circumcenter lies inside the triangle.
  • Right triangle: The circumcenter lies on the hypotenuse (at its midpoint).
  • Obtuse triangle: The circumcenter lies outside the triangle.

Finding the Circumcenter from Vertices

When given three vertices (x1, y1), (x2, y2), and (x3, y3), the circumcenter can be found by solving the system of equations formed by the perpendicular bisectors of any two sides. The circumradius can also be computed by first calculating the side lengths using the distance formula, then applying R = abc / (4K).

Practical Applications

  • Computer graphics: Circumscribed circles are used in Delaunay triangulation, a fundamental algorithm in computational geometry.
  • Navigation: Determining the circle passing through three known points helps in triangulation-based positioning.
  • Engineering: Fitting a circle through three points on a curved surface for quality control measurements.
  • Architecture: Designing circular arches or domes that pass through specific support points.
  • Astronomy: Determining orbital paths from three observed positions of a celestial body.

Relationship to Other Triangle Circles

Every triangle has four commonly studied circles: the circumscribed circle (circumcircle), the inscribed circle (incircle), and the three excircles. The circumradius R and inradius r are related by Euler's formula: OI2 = R2 - 2Rr, where O is the circumcenter and I is the incenter.

Tips for Accurate Calculations

  • Verify that the three sides satisfy the triangle inequality (sum of any two sides must exceed the third) before computing.
  • When using vertices, ensure the three points are not collinear (they must form an actual triangle).
  • For numerical stability with very obtuse or very flat triangles, use extended precision arithmetic.
  • Remember that the circumradius is always positive for a valid triangle.