To solve this problem, we need to recall a key property of regular polygons:

The sum of the exterior angles of any polygon is always $360^\circ$.

Since the octagon is regular, all the exterior angles are equal. This means:
$\text{Exterior angle} = \frac{\text{Sum of exterior angles}}{\text{Number of sides}}.$

For a regular octagon ($n = 8$):
$v = \frac{360^\circ}{8} = 45^\circ.$

Thus, the value of $v$ is $45^\circ$.

Would you like me to explain further or go into any specific detail?

Here are 5 related questions for exploration:

1. What is the formula for finding the exterior angle of a regular polygon?
2. How do interior and exterior angles of a polygon relate to each other?
3. What is the sum of the interior angles of a regular octagon?
4. If the shape were a hexagon instead, what would the value of $v$ be?
5. How can you confirm the sum of all $v$-angles equals $360^\circ$ for any polygon?

Tip: For regular polygons, always start with the fact that exterior angles sum up to $360^\circ$, no matter the number of sides.