1. **State the problem:** We are given a circle with center $O$ and triangle $ABC$ circumscribed about the circle such that $\angle A = 46^\circ$. We are asked to find the measure of $\angle O$, the angle at the center $O$ between lines $OC$ and $OB$. 2. **Understand the setup:** $O$ is the center of the circle. Points $B$ and $C$ lie on the circle. $\angle A$ is an inscribed angle subtending arc $BC$ of the circle. 3. **Recall a key fact:** The measure of the central angle (angle at $O$) subtending an arc is twice the measure of the inscribed angle subtending the same arc. 4. **Apply the fact:** Since $\angle A = 46^\circ$ is the inscribed angle subtending arc $BC$, the central angle $\angle BOC$ (labeled as $\angle O$) is $$\angle O = 2 \times \angle A = 2 \times 46^\circ = 92^\circ.$$ 5. **Final answer:** $$\boxed{92^\circ}.$$