To find the side length $x$ of a regular octagon, we can use the relationship between the side length and the radius $R$ of the circumscribed circle (the circle that passes through all vertices of the octagon).

Given that the octagon is regular, we can derive the side length using the following formula:

$x = R \cdot \sqrt{2 - 2\cos\left(\frac{360^\circ}{8}\right)}$

Simplifying the cosine term:

$\cos\left(\frac{360^\circ}{8}\right) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$

Thus, the formula for the side length becomes:

$x = R \cdot \sqrt{2 - 2 \cdot \frac{\sqrt{2}}{2}} = R \cdot \sqrt{2 - \sqrt{2}}$

If we do not know the radius $R$, and only have the measurement directly, we need additional information or measurements.

Would you be able to provide the radius or any other relevant measurements?

Here are some related questions that may help clarify and expand on this topic:

1. How do you find the radius of the circumscribed circle of a regular octagon given its side length?
2. How can the apothem (the perpendicular distance from the center to a side) be used to find the side length of a regular octagon?
3. What is the formula for the area of a regular octagon in terms of its side length?
4. How do you derive the side length formula of a regular octagon using trigonometry?
5. Can you determine the side length of a regular octagon if you know its perimeter?
6. How does the side length of a regular octagon relate to the side length of an inscribed square?
7. What are some real-life applications where knowing the side length of an octagon is useful?
8. How can the properties of symmetry in a regular octagon be used to simplify calculations?

Tip: For regular polygons, breaking them down into triangles can often simplify the process of finding side lengths and areas.