Circumscribed Circle Calculator
Triangle circumcircle calculator: circumradius R, diameter, circle area, and circumcenter — from 3 sides or 3 coordinate points.
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Results
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Verified Examples
Methodology
From three sides a, b, c: semiperimeter s = (a+b+c)/2, triangle area via Heron’s formula K = √(s(s−a)(s−b)(s−c)), then circumradius R = abc / 4K.
From three points: the circumcenter is the intersection of the perpendicular bisectors, computed with a determinant; R is the distance from the circumcenter to each vertex. Right triangle: the circumcenter lies exactly at the hypotenuse’s midpoint (3-4-5 example above: (2, 1.5), R = 2.5 = half the hypotenuse).
Validation: sides must be > 0 and satisfy the triangle inequality (a+b>c, etc.); the three points must not be collinear. See also the triangle area calculator.
FAQ
What is a triangle’s circumcircle?
The circumcircle is the circle passing through all three vertices of a triangle. Its center is called the circumcenter and its radius the circumradius (R). Every non-degenerate triangle has exactly one circumcircle.
What is the circumradius formula from three sides?
R = abc / 4K, where K is the triangle’s area computed via Heron’s formula: s = (a+b+c)/2, then K = sqrt(s(s-a)(s-b)(s-c)). Example: sides 3, 4, 5 give s = 6, K = 6, so R = 60/24 = 2.5.
How do I find the circumcenter from three coordinate points?
The circumcenter is the intersection of the three perpendicular bisectors of the triangle’s sides. This calculator uses the determinant formula: with D = 2(x1(y2-y3) + x2(y3-y1) + x3(y1-y2)), the center coordinates are ux = ((x1^2+y1^2)(y2-y3) + (x2^2+y2^2)(y3-y1) + (x3^2+y3^2)(y1-y2)) / D and analogously uy. The radius R is then the distance from the center to any vertex.
When does the calculator reject my input?
Three conditions are rejected with an error message: (1) any side is zero or negative; (2) the sides violate the triangle inequality, e.g. 1, 2, 3 (1+2 is not greater than 3, a degenerate triangle); (3) the three points are collinear, so there is no unique circumcircle.
How does R relate to the circle’s diameter and area?
Diameter = 2R and circumcircle area = pi x R^2. For the 3-4-5 triangle (R = 2.5): diameter = 5 and circle area = 19.634954.
Where is the circumcenter for right, acute, and obtuse triangles?
Right triangle: the circumcenter is exactly at the hypotenuse’s midpoint (R = half the hypotenuse). Acute triangle: the circumcenter is inside the triangle. Obtuse triangle: the circumcenter is outside the triangle. Example: the 3-4-5 right triangle at points (0,0)-(4,0)-(0,3) has circumcenter (2, 1.5).
Is R = a / (2 sin A) really true?
Yes, that is the law of sines: a / sin A = b / sin B = c / sin C = 2R. This formula is equivalent to R = abc / 4K because K = (1/2)bc sin A. This calculator uses the abc/4K form so that sides or coordinates alone suffice.
What is the difference between circumradius and inradius?
Circumradius (R) is the radius of the OUTER circle passing through all three vertices; inradius (r) is the radius of the INNER circle tangent to all three sides, with r = K/s. For the 3-4-5 triangle: R = 2.5 while r = 1. This calculator shows both.
Free tool by NRTechStudio