Circumscribed Circle (Circumcircle) Calculator

Calculate the circumscribed circle (circumcircle) of a triangle. Enter three sides or three vertex coordinates to find the circumradius, circumcenter, area, angles, and see an interactive diagram with step-by-step formulas.

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About Circumscribed Circle (Circumcircle) Calculator

The Circumscribed Circle (Circumcircle) Calculator finds the circumscribed circle of any triangle. The circumscribed circle — also known as the circumcircle — is the unique circle that passes through all three vertices of a triangle. Enter three side lengths or three vertex coordinates to instantly compute the circumradius, circumcenter location, triangle area, interior angles, and more, with an interactive SVG diagram and step-by-step formulas.

Key Concepts of the Circumscribed Circle

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Circumcircle

The unique circle passing through all 3 vertices of a triangle

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Circumcenter

The center point, equidistant from all 3 vertices

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Perpendicular Bisectors

Three lines that meet at the circumcenter

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Circumradius

R = abc / (4K), where K is the triangle area

Circumscribed Circle Formulas

For a triangle with sides a, b, c and semi-perimeter s = (a + b + c) / 2:

Property	Formula	Description
Triangle Area (Heron's)	\(K = \sqrt{s(s-a)(s-b)(s-c)}\)	Area from three sides using semi-perimeter
Circumradius	\(R = \frac{abc}{4K}\)	Radius of the circumscribed circle
Circumcircle Area	\(A = \pi R^2\)	Area enclosed by the circumcircle
Circumference	\(C = 2\pi R\)	Perimeter of the circumcircle
Inradius	\(r = \frac{K}{s}\)	Radius of the inscribed circle
Euler's Distance	\(d = \sqrt{R(R-2r)}\)	Distance between circumcenter and incenter

Circumcenter Location by Triangle Type

The position of the circumcenter depends on the type of triangle:

Acute triangle: The circumcenter lies inside the triangle. All angles are less than 90°, so the perpendicular bisectors intersect within the triangle's interior.
Right triangle: The circumcenter lies exactly at the midpoint of the hypotenuse. The circumradius equals half the hypotenuse length.
Obtuse triangle: The circumcenter lies outside the triangle, on the opposite side from the obtuse angle. This is because the perpendicular bisectors diverge outward.

How to Find the Circumscribed Circle

Choose your input method: Select "Three Sides" if you know the side lengths a, b, c, or "Three Vertices" if you have the coordinates of each vertex.
Enter the values: Input the three side lengths or the (x, y) coordinates of vertices A, B, and C. Click a quick example to auto-fill sample values.
Click Calculate: Press the "Calculate Circumscribed Circle" button.
Review the results: See the circumradius R, circumcenter coordinates, circumcircle area and circumference, triangle area, angles, inradius, and the R/r ratio.
Explore the diagram: Toggle overlays for the circumcircle, perpendicular bisectors, radii, incircle, and labels to visualize the geometry.

Practical Applications

The circumscribed circle has important applications in many fields. In surveying and navigation, the circumcircle helps determine positions using triangulation. In computer graphics, Delaunay triangulation maximizes minimum angles by ensuring no vertex lies inside any triangle's circumcircle. In engineering, circumscribed circles define minimum enclosing boundaries for triangular components. The circumcircle is also fundamental in computational geometry algorithms for mesh generation and Voronoi diagrams.

Euler's Theorem and the Circumcircle

Euler's inequality states that for any triangle, the circumradius R is at least twice the inradius r: R ≥ 2r. Equality holds only for equilateral triangles. Additionally, Euler's formula relates the distance d between the circumcenter O and incenter I as \(d^2 = R(R - 2r)\). This elegant result connects the two most fundamental circles associated with a triangle and reveals deep properties of triangle geometry.

FAQ

What is a circumscribed circle (circumcircle)?

A circumscribed circle, or circumcircle, is the unique circle that passes through all three vertices of a triangle. Its center is called the circumcenter, and its radius is the circumradius. Every non-degenerate triangle has exactly one circumscribed circle.

How do you find the circumradius of a triangle?

The circumradius R of a triangle with sides a, b, c and area K is R = abc / (4K). First find the area using Heron's formula: K = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2. Then divide the product of the three sides by four times the area.

Where is the circumcenter of a triangle?

The circumcenter is the intersection point of the three perpendicular bisectors of the triangle's sides. For acute triangles, it lies inside the triangle. For right triangles, it sits at the midpoint of the hypotenuse. For obtuse triangles, it lies outside the triangle, beyond the longest side.

Does every triangle have a circumscribed circle?

Yes, every non-degenerate triangle (three non-collinear points) has exactly one circumscribed circle. This is a fundamental theorem of Euclidean geometry: three non-collinear points uniquely determine a circle.

What is the relationship between circumradius and inradius?

Euler's inequality states R ≥ 2r for any triangle, where R is the circumradius and r is the inradius. Equality holds only for equilateral triangles. Euler's formula also gives the distance between the two centers: d² = R(R − 2r), where d is the distance between circumcenter and incenter.

Reference this content, page, or tool as:

"Circumscribed Circle (Circumcircle) Calculator" at https://MiniWebtool.com/circumscribed-circle-circumcircle-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-03

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