What does circumcentre mean?
A circumcentre is the unique point at which the three perpendicular bisectors of a triangle's sides intersect. Because each bisector marks points equidistant from two vertices, the circumcentre lies at exactly the same distance from all three corners of the triangle, and it therefore serves as the centre of the circle that passes through them — the so-called circumcircle. Its position depends on the shape of the triangle: in an acute triangle it falls inside, in a right-angled triangle it coincides with the midpoint of the hypotenuse, and in an obtuse triangle it lies outside the figure altogether. The term is chiefly encountered in school geometry, olympiad mathematics, and technical writing, appearing most often in phrases such as 'the circumcentre of a triangle'. British English spells it with '-re', while American English uses '-er'. Related centres include the centroid, incentre, and orthocentre.
nounThe point at which the perpendicular bisectors of the sides of a triangle meet; it is equidistant from all three vertices and is the centre of the triangle's circumscribed circle (circumcircle).
- The common point of the perpendicular bisectors of the sides of a triangle, which is the centre of its circumscribed circle.
- By extension, the corresponding point for any cyclic polygon or for a circle passing through given points.
"The circumcentre of an acute-angled triangle always lies inside the triangle itself."
"Using coordinate geometry, we found the circumcentre of the triangle to lie at the origin."
"Because the triangle was obtuse, its circumcentre fell outside the figure, surprising several students in the class."
The plural is regular and typically appears when comparing triangles or discussing multiple figures.
"Different triangles can have their circumcentres in very different positions relative to their vertices."
Every triangle has one — and depending on whether it's acute, right-angled, or obtuse, that single point can sit inside, exactly on, or entirely outside the triangle.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of circumcentre
The word combines the Latin prefix 'circum-' meaning 'around' with 'centre', ultimately from Latin 'centrum', itself borrowed from Greek 'kentron' meaning 'sharp point' or 'pivot'. The compound reflects the point's role as the centre of a circle drawn around a figure. Such geometry terms were coined in the modern era as European mathematicians systematised classical Greek geometry, and cognate forms appear across European languages, including French 'circoncentre' and German 'Umkreismittelpunkt' by different construction.
Related word forms
How circumcentre is actually used
Primarily used in geometry and mathematics education. The British spelling is 'circumcentre'; American usage typically favours 'circumcenter'. It is a technical term with no figurative or everyday use.
Easily confused with circumcentre
The centroid is the intersection of a triangle's medians (its geometric 'average' of vertices), while the circumcentre is the intersection of the perpendicular bisectors of its sides.
The orthocentre is where the altitudes of a triangle meet, whereas the circumcentre is where the perpendicular bisectors of the sides meet.
The incentre is the meeting point of the angle bisectors, equidistant from the sides, while the circumcentre is the meeting point of the side bisectors, equidistant from the vertices.