Circumscribed Circle Calculator
Our free triangle calculator solves circumscribed circle problems. Get worked examples, visual aids, and downloadable results.
Formula
R = (a x b x c) / (4 x Area) or R = a / (2 sin A)
The circumradius R can be found from three sides using Herons formula for the area, or from one side and its opposite angle using the law of sines. The circumscribed circle passes through all three vertices of the triangle.
Worked Examples
Example 1: Circumscribed Circle of a Scalene Triangle
Problem: Find the circumscribed circle of a triangle with sides 5, 7, and 9.
Solution: Semi-perimeter s = (5 + 7 + 9) / 2 = 10.5\nArea = sqrt(10.5 x 5.5 x 3.5 x 1.5) = sqrt(303.1875) = 17.4123\nCircumradius R = (5 x 7 x 9) / (4 x 17.4123) = 315 / 69.6493 = 4.5227\nDiameter = 9.0454\nCircle circumference = 2 x pi x 4.5227 = 28.4148\nCircle area = pi x 4.5227^2 = 64.2649
Result: R = 4.5227 | Diameter = 9.0454 | Circle Area = 64.2649 | Triangle Area = 17.4123
Example 2: Circumscribed Circle Using Side and Angle
Problem: A triangle has a side of 12 cm opposite an angle of 50 degrees. Find the circumscribed circle.
Solution: Using the law of sines: R = side / (2 x sin(angle))\nR = 12 / (2 x sin(50)) = 12 / (2 x 0.7660) = 12 / 1.5321 = 7.8318\nDiameter = 15.6636 cm\nCircumference = 2 x pi x 7.8318 = 49.2083 cm\nCircle area = pi x 7.8318^2 = 192.7455 sq cm
Result: R = 7.8318 cm | Diameter = 15.6636 cm | Circle Area = 192.7455 sq cm
Frequently Asked Questions
How do you find the circumscribed circle using three sides?
To find the circumscribed circle from three sides a, b, c, first compute the semi-perimeter s = (a+b+c)/2 and the triangle area using Herons formula: Area = sqrt(s(s-a)(s-b)(s-c)). Then apply the circumradius formula R = (abc)/(4 times Area). The circumference of the circumscribed circle is 2 times pi times R, and its area is pi times R squared. For example, a triangle with sides 3, 4, 5 has s = 6, Area = 6, and R = (3 times 4 times 5)/(4 times 6) = 60/24 = 2.5. The circumscribed circle has diameter 5 (the hypotenuse, confirming this is a right triangle).
What is the relationship between circumscribed and inscribed circles?
Every triangle has both a circumscribed circle (circumcircle, passing through vertices) and an inscribed circle (incircle, tangent to all three sides). The circumradius R is always greater than or equal to twice the inradius r, with equality only for equilateral triangles. Euler proved that the distance d between the circumcenter and incenter satisfies d squared = R(R - 2r), known as Eulers formula. The ratio of the circumscribed circle area to the inscribed circle area equals (R/r) squared, and this ratio is minimized at 4 for equilateral triangles.
How does the law of sines connect to the circumscribed circle?
The extended law of sines states that a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. This means that the ratio of any side to the sine of its opposite angle equals the diameter of the circumscribed circle. This provides an elegant alternative formula for finding R: just divide any side by twice the sine of its opposite angle. The law of sines essentially encodes the circumscribed circle into the fundamental relationship between sides and angles, making the circumradius a central quantity in triangle trigonometry.
Can you construct a circumscribed circle with compass and straightedge?
Yes, constructing a circumscribed circle with compass and straightedge is a classic geometric construction. First, draw perpendicular bisectors of any two sides of the triangle (the third bisector will pass through the same point). The intersection of these perpendicular bisectors is the circumcenter. Then, set your compass radius to the distance from the circumcenter to any vertex and draw the circle. This construction works because every point on a perpendicular bisector of a segment is equidistant from both endpoints, so the intersection point is equidistant from all three vertices.
What is the circumscribed circle of an equilateral triangle?
For an equilateral triangle with side length s, the circumradius equals s times sqrt(3) / 3, or equivalently s / sqrt(3). The circumcenter coincides with the centroid, incenter, and orthocenter since all triangle centers merge for equilateral triangles. The circumscribed circle area is pi times s squared / 3, and the ratio of circumscribed circle area to triangle area is (4 pi) / (3 sqrt(3)), approximately 2.418. The inradius is exactly half the circumradius (R = 2r), which is the minimum possible ratio for any triangle and confirms the equilateral triangle is the most symmetric.
How is the circumscribed circle used in Delaunay triangulation?
Delaunay triangulation is a fundamental algorithm in computational geometry that relies heavily on circumscribed circles. The key property of a Delaunay triangulation is that no point in the dataset lies inside the circumscribed circle of any triangle in the triangulation. This maximizes the minimum angle among all possible triangulations, avoiding very skinny triangles. Delaunay triangulation is widely used in mesh generation for finite element analysis, terrain modeling from scattered elevation data, nearest-neighbor interpolation, and computer graphics. The circumscribed circle test (checking if a point lies inside a circumcircle) is the core operation.