Angle at a point

Definition

Let $A$, $B$ and $C$ be points of a Euclidean space $\mathbb{R}^n$ with $A \neq B$ and $C \neq B$. The angle $\angle ABC$ is the angle between the vectors $A - B$ and $C - B$. Thus it is the unique real number $\theta$ with $0 \leq \theta \leq \pi$ such that $$\cos \theta = \frac{(A - B) \cdot (C - B)}{\|A - B\| \, \|C - B\|}.$$ The point $B$ is the vertex of the angle.

Remarks

The angle is symmetric: $\angle ABC = \angle CBA$. The angle is $\pi$ if and only if $B$ lies strictly between $A$ and $C$ on a line. It is $0$ if and only if $A$ and $C$ are on the same ray from $B$. Mathlib's EuclideanGeometry.angle uses the same formula for all points. It gives $\pi / 2$ when $A = B$ or $C = B$. The angle is a right angle when $\theta = \pi / 2$, acute when $\theta < \pi / 2$ and obtuse when $\theta > \pi / 2$.

Explanations

L1 human

To measure the angle at $B$, stand at $B$ and look at $A$, then turn to look at $C$. The angle is how far you turn, using the short way round. The order of $A$ and $C$ does not matter.