Bond angle ($\theta$)

A bond angle is the angle between two chemical bonds sharing a central chemical atom. In H–O–H, the middle symbol identifies the vertex: the angle is at oxygen. Let $u$ and $v$ be the nonzero displacement vectors from that atomic nucleus to the two neighbours. The actual three-dimensional bond angle satisfiesusing the dot product and Euclidean norm. Rotating the whole molecule preserves this expression; projecting it onto a page generally does not.

There is another useful way to find a bond angle. If the two bond lengths are $a,b$ and the distance between the outer atomic nuclei is $d$, the law of cosines givesFor equal bond lengths $a=b$, this becomes $d=2b\sin (\theta /2)$. Thus a measured outer-atom separation contains information about a bond angle even though those outer chemical atoms are not directly joined by a chemical bond.

For a stipulated rigid, equal-bond length model using water's $104.4776^{\circ }$ equilibrium bond angle, $d/b\approx 1.5811$, whereas a straight H–O–H arrangement would give $d/b=2$. These are calculated geometric ratios, not new measured distances. Gas electron diffraction illustrates how pair-distance information can constrain molecular geometry. Substituting independently averaged distances into the rigid triangle formula does not automatically recover an equilibrium bond angle, because averaging and this nonlinear calculation need not commute.