1. **Problem statement:** We have a regular octagon ABCDEFGH with center O. We need to calculate the size of angle $AOB$, where $A$ and $B$ are adjacent vertices and $O$ is the center. 2. **Key fact:** In a regular polygon with $n$ sides, the central angle between two adjacent vertices is given by the formula: $$\text{Central angle} = \frac{360^\circ}{n}$$ 3. Since the octagon has $n=8$ sides, the central angle between adjacent vertices $A$ and $B$ is: $$\angle AOB = \frac{360^\circ}{8}$$ 4. Calculate the value: $$\angle AOB = 45^\circ$$ 5. **Explanation:** The center $O$ connects to each vertex, dividing the octagon into 8 equal isosceles triangles. The angle at the center for each triangle is $45^\circ$, so the angle $AOB$ formed by radii $OA$ and $OB$ is $45^\circ$. **Final answer:** $$\boxed{45^\circ}$$