= Bond angle
{title2=$\theta$}
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> satisfies
$$
\cos\theta=\frac{u\cdot v}{\|u\|\|v\|},\qquad 0\leq\theta\leq\pi,
$$
using 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> gives
$$
d^2=a^2+b^2-2ab\cos\theta.
$$
For 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>, $d/b\approx1.5811$, 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.
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