# What is the correct mathematical language in French to distinguish between vertex and angle?

I was presented on Facebook by a French puzzle from one of my French relatives as follows:

Combien d'angles possède un cube?

With 2, 4, 6 or 8, as the possible choices. Now I know there are 24 angles in a cube, but 8 vertices. Google translator is telling me that l'angle translates to corner as well as angle. Now, in English, vertices can be called corners, but the latter is less formal. So what are the terms used by French mathematicians to refer to vertices and angles?

Addendum: Having now read several comments, I need to be clear that in English, vertex is the main word used for the intersection of lines/sides in a polygon (2D figure) or lines/edges in a polyhedron. And, to complicate matters, with a polyhedron in 3D, 3 planes can intersect to form a vertex, not just three lines. But, and this is important, in any dimension, linear objects are points, lines, planes and thereafter, for linear objects of dimension greater than 2, hyperplanes. In all dimensions, including and beyond 2, angles are the orientation between lines, planes and hyperplanes in various combinations of all three.

My specific question is, what are the correct French mathematical term for the specific English mathematical terms, vertex and angle? It's not about everyday uses or confusions of those terms, which are legion.

• Neither of these choices are right. A cube only has one angle: the angle droit! ;-) Commented Mar 16 at 18:09
• @jlliagre Mais tu vois, on ne dit pas trop qu'un cube a un angle. C'est pas clair en 3D. On dirait plutôt qu'il y a un angle entre deux plans qui contiennent 2 faces du cube, par exemple, ou directement entre deux faces du cube. Commented Mar 16 at 23:05
• @Frank Ne prends pas trop au sérieux mon commentaire, mais tous les angles qu'on peut mesurer sur un cube sont quand même droits... Commented Mar 16 at 23:20
• Please take it from an English mathematician using English maths language that there are 12 edges, 8 vertices, 6 faces and 24 angles in a cube. It could, perhaps, be argued that there are more than 24 angles in a cube because there are 24 between unique pairs of intersecting lines, but also 12 between the 12 unique pairs of intersecting faces, but it starts to get counter productive. A cube can be defined as simply as the twelve edges produced by the intersection of three, equidistant, mutually perpendicular, parallel pairs of planes in 3D linear space. Commented Mar 17 at 7:39
• I won't answer the question since I have no mathematical knowledge, but I'd like to point out from a very layman's perspective that the French word angle can indeed be synonym of coin (corner) in, for example, le coin de la pièce (the corner of the room), and if anyone asked me Combien d'angles possède un cube? I'd answer 24 straight away. On the other hand I'd say un cube a 8 sommets and un cube a 6 faces.
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Commented Mar 17 at 8:46

The answer to your revised question is contained in the comments, but since you still seem unsatisfied, I will summarize them in an answer.

The word for a vertex in French is un sommet. This is the correct translation both in graph theory (where a graph has vertices and edges), for two-dimensional polygons, and for three-dimensional polyhedra.

In mathematical English, the word angle is not generally used for edges of a cube. Polygons have vertices and sides (or edges). Polyhedra have vertices, edges and faces (or facets).

In French, an edge of a polygon is called un coté. An edge of a polyhedron is called une arête. An edge of a graph is also called une arête.

The use of the word angle as the intersection of two rays, or as a measure of such an intersection (in radians or degrees) is translated to French as un angle. For example:

The sum of the angles of a triangle is 180 degrees.
La somme des angles d'un triangle est égale à 180°.

The intersection of two planes in three dimensions also forms an angle; for example, two perpendicular planes intersect at an angle of 90°. This usage of angle is also translated as angle in French.

• The definitions and translations in Termium Plus, (filtering → mathematics →plane geometry) are reliable, they seem to concur with what you say.
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Commented Mar 17 at 14:16
• Unsatisfied is the wrong word. I would like to be able to carry out the usual Stack Exchange protocol. My addendum was merely to add context for people who aren't mathematicians. I didn't change my original question at all. I will ultimately accept the answer that I think has best answered my original question. Commented Mar 18 at 11:30
• I don't think angles (24 in a cube) are mathematically a very important feature of polyhedra, and I wouldn't be surprised if French mathematicians don't have a word for them. Commented Mar 18 at 12:02

This sentence does not sound like mathematical language, but I guess facebook quizz are not made by Math teachers.

One would expect something like :

Combien de sommets possède un cube ?

• You might add that angle is indeed the correct translation for angle for mathematics – at least for some meanings of angle – although according to wikipedia, a cube has douze arêtes and not douze angles. Commented Mar 16 at 12:57
• @PeterShor An angle would be about 2 lines in a plane. We can say l'angle entre deux plans, but the intersection of 2 planes would be an arête in a geometrical object like a cube. Otherwise, the intersection of 2 planes in general is just la droite formée par l'intersection de deux plans. Commented Mar 16 at 14:53
• Also, arête does not seem to work for planar figures. We don't say une arête d'un triangle, but un côté d'un triangle. Arête seems to be specific to 3D. Commented Mar 16 at 17:36
• @Frank: That makes sense — an arête is a ridge, and a sommet is a peak. So if anything's surprising, it's that sommet is used for vertices in two dimensions. Commented Mar 16 at 18:26
• In English in 2D polygons you have sides, vertices and angles and in 3D polyhedrons you have edges, faces, vertices and angles. Commented Mar 17 at 6:52

What is the correct mathematical language in French to distinguish between vertex and angle?

A vertex (sommet or just point) and an angle (angle) are never confused in French.

My guess is then that the quiz was talking about solid angles (angles solides) with which the answer '8' is valid.

A cube has indeed 8 vertices but from each of them also 8 solid angles encompassing one eighth of the space: 4𝜋/8 = 𝜋/2 steradians.

• Oui, l'angle solide vient à l'esprit, puisqu'un cube est un objet dans un espace à 3 dimensions. Mais je ne sais pas si c'est cela qu'on recherche ici. Commented Mar 17 at 15:37
• @Frank Disons que c'est une réponse compatible avec l'intitulé de la question du quiz et les choix qu'il propose alors que deux, quatre et six sont plus difficiles à envisager. Commented Mar 17 at 15:54
• Il n'y a que moi pour te plussoyer? Les angles solides ne sont pas très populaires... :-) En plus, les angles solides ont le mérite de mesurer pi/2, qui est la même valeur qu'un angle droit dans le plan - très satisfaisant. Commented Mar 17 at 19:14
• @Frank Le stéradian n'attire pas foule ;-) C'est pourtant une manière de résoudre le postulat incorrect de l'OP. Commented Mar 17 at 19:37
• Effectivement, ça pourrait être ça. Mais est-ce habituel d'utiliser "angle" tout court pour parler d'angle solide ? Commented Mar 18 at 9:33