Circumscribed Circle Calculator
Our free triangle calculator solves circumscribed circle problems. Get worked examples, visual aids, and downloadable results.
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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.
Last reviewed: December 2025
Worked Examples
Example 1: Circumscribed Circle of a Scalene Triangle
Example 2: Circumscribed Circle Using Side and Angle
Background & Theory
The Circumscribed Circle Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Circumscribed Circle Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy