Calculate the circumscribed circle of a triangle from its sides. Find the circumradius, circumcenter coordinates, circumcircle area, Euler line distance, and area ratios. Includes presets and...
The circumscribed circle (circumcircle) of a triangle is the unique circle that passes through all three vertices. Its center is called the circumcenter, and its radius is the circumradius (R). The circumcenter is equidistant from all three vertices and is found at the intersection of the perpendicular bisectors of the sides.
The circumradius is computed from the extended law of sines: R = abc / (4·Area), or equivalently R = a / (2·sin A). This elegant formula connects the circumcircle to both the side lengths and the angles of the triangle.
The circumcenter's position depends on the triangle type. For acute triangles, it lies inside. For right triangles, it sits at the midpoint of the hypotenuse. For obtuse triangles, it falls outside the triangle. This geometric fact has practical implications in surveying and GPS triangulation.
The Euler line is a remarkable line in any non-equilateral triangle that passes through the circumcenter (O), centroid (G), and orthocenter (H). The centroid divides the segment OH in a 2 : 1 ratio: OG = ⅓ OH.
This calculator computes the circumradius, circumcircle area, circumcenter coordinates, triangle area, the ratio of circumcircle-to-triangle areas, and the Euler line distance. Presets cover classic triangles, and the reference table shows how circumradius varies across common triangle types.
This calculator is useful when you need more than the textbook circumradius formula. It shows where the circumcenter actually sits, how large the full circumcircle is compared with the triangle, and how the Euler line changes as the triangle shifts from acute to right to obtuse. That makes it valuable for geometry proofs, coordinate setups, surveying layouts, and any problem where a circle through all three vertices matters as much as the radius itself.
Circumradius: R = abc / (4·Area) Extended law of sines: R = a / (2·sin A) Circumcircle area: A_circ = πR² Area ratio: A_circ / A_tri = πR² / Area Euler line: OH² = R² − 8R²·cos A·cos B·cos C Euler relation: OG = ⅓ OH (centroid divides OH in 2:1)
Result: R = 2.50, Circumcircle area ≈ 19.63, Triangle area = 6, Ratio ≈ 3.27
For a 3-4-5 right triangle: Area = 6, R = 3·4·5 / (4·6) = 60/24 = 2.5. The circumcenter is at the midpoint of the hypotenuse. Circumcircle area = π·2.5² ≈ 19.63.
One of the most useful things about a circumcircle calculator is that it makes the circumcenter tangible instead of abstract. In an acute triangle the center stays inside, in a right triangle it lands on the midpoint of the hypotenuse, and in an obtuse triangle it moves outside the figure entirely. Seeing that position alongside the radius helps you connect the algebraic answer to the actual geometry of the triangle.
The circumradius by itself can be hard to interpret until you compare it with the triangle area and the full circle area. A small change in side lengths can produce a much larger circumcircle when the triangle becomes flatter or more obtuse. That is why the area ratio in this calculator matters: it tells you how much larger the vertex-passing circle is than the triangle it contains, which is useful in layout, drafting, and proof work.
For non-equilateral triangles, the circumcenter, centroid, and orthocenter line up on the Euler line. When the calculator reports those points and distances, you can verify classic geometry relationships without building the construction by hand. This is especially helpful in coordinate geometry problems where you want to move from side lengths to triangle centers and check whether a derived diagram is behaving as expected.
The circumscribed circle (circumcircle) is the unique circle passing through all three vertices of a triangle. Its center is the circumcenter, and its radius is the circumradius.
Use R = abc / (4·Area), where a, b, c are the sides and Area is computed via Heron's formula. Alternatively, R = a / (2·sin A) from the law of sines.
Inside for acute triangles, at the midpoint of the hypotenuse for right triangles, and outside for obtuse triangles. Use this as a practical reminder before finalizing the result.
The Euler line is a line passing through the circumcenter, centroid, and orthocenter of a non-equilateral triangle. The centroid divides the circumcenter-orthocenter segment in a 2 : 1 ratio.
Euler's formula: d² = R(R − 2r), where d is the distance between circumcenter and incenter, R is the circumradius, and r is the inradius. This requires R ≥ 2r.
Yes. Every triangle (acute, right, or obtuse) has exactly one circumscribed circle passing through all three vertices.