# Circumscribed circle

A circumscribed circle (or circumcircle) of a triangle is the unique circle that passes through all three of the triangle's vertices. Its center is called the circumcenter and its radius the circumradius.<sup>[1](https://mathworld.wolfram.com/Circumcircle.html)</sup> More generally, a circle passing through every vertex of a polygon makes that polygon a cyclic polygon, and the circle is circumscribed about it.<sup>[2](https://mathworld.wolfram.com/CyclicPolygon.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | The unique circle through the three vertices of a triangle; center = circumcenter, radius = circumradius<sup>[1](https://mathworld.wolfram.com/Circumcircle.html)</sup> |
| Existence and uniqueness | Exactly one circle is circumscribed about any triangle; any three non-collinear points determine exactly one circle<sup>[3](https://math.stonybrook.edu/~scott/mat200.fall02/Geometry/Main/5Circles_lines.html)</sup> |
| Circumcenter location | Inside an acute triangle, at the midpoint of the hypotenuse of a right triangle, outside an obtuse triangle<sup>[4](https://mathworld.wolfram.com/Circumcenter.html)</sup> |
| Area formula | Triangle area \( K = \dfrac{abc}{4R} \), so \( R = \dfrac{abc}{4K} \)<sup>[5](https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_%28Corral%29/02%3A_General_Triangles/2.05%3A_Circumscribed_and_Inscribed_Circles)</sup> |
| Radius from sides | \( R = \dfrac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}} \), where \( s = \tfrac{1}{2}(a+b+c) \)<sup>[5](https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_%28Corral%29/02%3A_General_Triangles/2.05%3A_Circumscribed_and_Inscribed_Circles)</sup> |
| Law of sines | \( 2R = \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} \)<sup>[5](https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_%28Corral%29/02%3A_General_Triangles/2.05%3A_Circumscribed_and_Inscribed_Circles)</sup> |
| Right triangle | The hypotenuse is a diameter of the circumcircle<sup>[6](https://mathworld.wolfram.com/Circumradius.html)</sup> |

## Existence and uniqueness

Three non-collinear points determine exactly one circle. The Stony Brook geometry notes prove this in two parts: at most one circle can be circumscribed about a given triangle, since two distinct circles sharing the same three vertices would share the same center and radius and therefore be the same circle; and one always exists, because the perpendicular bisectors of the sides meet at a point equidistant from all three vertices.<sup>[3](https://math.stonybrook.edu/~scott/mat200.fall02/Geometry/Main/5Circles_lines.html)</sup> The same argument yields the corollary that the three perpendicular bisectors of any triangle's sides are concurrent.<sup>[3](https://math.stonybrook.edu/~scott/mat200.fall02/Geometry/Main/5Circles_lines.html)</sup>

## The circumcenter

The circumcenter \( O \) is the intersection of the three perpendicular bisectors of the sides.<sup>[1](https://mathworld.wolfram.com/Circumcircle.html)</sup> Each bisector is the locus of points equidistant from a side's two endpoints, so their common intersection is equidistant from all three vertices, and that common distance is the circumradius.<sup>[7](https://www.math.wichita.edu/~ryan/teaching/M621/syllabus/M621_text.pdf)</sup>

**Position relative to the triangle.** The circumcenter's location depends on the triangle's angles. It lies in the interior of an acute triangle, coincides with the midpoint of the hypotenuse in a right triangle, and lies outside an obtuse triangle.<sup>[4](https://mathworld.wolfram.com/Circumcenter.html)</sup> In the right-triangle case the hypotenuse itself is a diameter of the circumcircle, so the circumradius is half the hypotenuse.<sup>[6](https://mathworld.wolfram.com/Circumradius.html)</sup>

## Construction

To construct the circumcircle with compass and straightedge, find the midpoints of two sides, erect the perpendicular to each side at its midpoint, and mark their intersection \( O \). The circle centered at \( O \) through any vertex is the circumcircle, and the segment from \( O \) to a vertex is the circumradius.<sup>[7](https://www.math.wichita.edu/~ryan/teaching/M621/syllabus/M621_text.pdf)</sup> Two bisectors suffice because the third must pass through the same point.<sup>[7](https://www.math.wichita.edu/~ryan/teaching/M621/syllabus/M621_text.pdf)</sup>

## Formulas for the circumradius

**Area relation.** For a triangle with side lengths \( a \), \( b \), \( c \), angles \( A \), \( B \), \( C \), area \( K \), and circumradius \( R \),

\[ K = \frac{abc}{4R}, \qquad R = \frac{abc}{4K}. \]

The proof substitutes \( \sin A = a/(2R) \) and its cyclic analogs, which follow from the extended law of sines \( 2R = a/\sin A = b/\sin B = c/\sin C \), into the area formula \( K = a^2 \sin B \sin C / (2 \sin A) \).<sup>[5](https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_%28Corral%29/02%3A_General_Triangles/2.05%3A_Circumscribed_and_Inscribed_Circles)</sup>

**Radius from the three sides alone.** Combining the area relation with [Heron's formula](https://www.edgepedia.org/herons-formula) \( K = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s = \tfrac{1}{2}(a+b+c) \) is the semiperimeter, gives

\[ R = \frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}}. \]

This expresses the circumradius purely in terms of the side lengths.<sup>[5](https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_%28Corral%29/02%3A_General_Triangles/2.05%3A_Circumscribed_and_Inscribed_Circles)</sup>

## Circumcircles of polygons

A polygon whose vertices all lie on one circle is called cyclic, and the circle is its circumscribed circle.<sup>[2](https://mathworld.wolfram.com/CyclicPolygon.html)</sup> Every triangle is cyclic, but a general polygon need not be: four or more points lie on a common circle only under additional conditions.<sup>[2](https://mathworld.wolfram.com/CyclicPolygon.html)</sup> For any set of \( n \) side lengths that form a simple closed polygon, a cyclic \( n \)-gon exists for every \( n \ge 3 \), and among all \( n \)-gons with those side lengths the cyclic one has the largest possible area.<sup>[2](https://mathworld.wolfram.com/CyclicPolygon.html)</sup> A circumcircle of a polygon is the two-dimensional case of a circumsphere of a solid.<sup>[1](https://mathworld.wolfram.com/Circumcircle.html)</sup>

## Relation to the minimum enclosing circle

The circumscribed circle of a triangle is not, in general, the smallest circle containing the triangle. In computational geometry, the smallest enclosing circle of a set of points is unique and is either the circumcircle of three points of the set or the circle defined by two points as a diameter.<sup>[8](https://www.cs.rpi.edu/~cutler/classes/computationalgeometry/F23/references/Computational_Geometry_An_Introduction.pdf)</sup> When the triangle is acute or right, its circumcircle is the minimum enclosing circle of its vertices; when the triangle is obtuse, the minimum circle instead has the longest side as a diameter, which is consistent with the circumcenter lying outside the triangle.<sup>[4](https://mathworld.wolfram.com/Circumcenter.html)</sup><sup> • </sup><sup>[8](https://www.cs.rpi.edu/~cutler/classes/computationalgeometry/F23/references/Computational_Geometry_An_Introduction.pdf)</sup> The minimum enclosing circle can be computed in \( O(n) \) expected time by randomized incremental construction, and the enclosing-circle problem is known in operations research as a minimax facilities location problem.<sup>[9](https://yunus.hacettepe.edu.tr/~burkay.genc/courses/bca608_compgeo/slides/slides4b.pdf)</sup><sup> • </sup><sup>[8](https://www.cs.rpi.edu/~cutler/classes/computationalgeometry/F23/references/Computational_Geometry_An_Introduction.pdf)</sup>

## References

1. [Circumcircle, Wolfram MathWorld](https://mathworld.wolfram.com/Circumcircle.html)
2. [Cyclic Polygon, Wolfram MathWorld](https://mathworld.wolfram.com/CyclicPolygon.html)
3. [5 Circles and Lines, MAT 200 course notes, Stony Brook University](https://math.stonybrook.edu/~scott/mat200.fall02/Geometry/Main/5Circles_lines.html)
4. [Circumcenter, Wolfram MathWorld](https://mathworld.wolfram.com/Circumcenter.html)
5. [2.5: Circumscribed and Inscribed Circles, Elementary Trigonometry (Corral), Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_%28Corral%29/02%3A_General_Triangles/2.05%3A_Circumscribed_and_Inscribed_Circles)
6. [Circumradius, Wolfram MathWorld](https://mathworld.wolfram.com/Circumradius.html)
7. [Euclid and Hilbert (course text), Wichita State University](https://www.math.wichita.edu/~ryan/teaching/M621/syllabus/M621_text.pdf)
8. [Preparata, F. P., and Shamos, M. I. Computational Geometry: An Introduction, Springer](https://www.cs.rpi.edu/~cutler/classes/computationalgeometry/F23/references/Computational_Geometry_An_Introduction.pdf)
9. [Lecture 6: Smallest enclosing circles and more, Hacettepe University](https://yunus.hacettepe.edu.tr/~burkay.genc/courses/bca608_compgeo/slides/slides4b.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Elementary and Euclidean geometry › Plane figures*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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