To solve for $x$ in the given circle geometry diagram:

1. Understand the setup:

   • $O$ is the center of the circle.
   • $x^\circ$ is the angle at the center ($\angle O$).
   • The angle at the circumference is $40^\circ$.

2. Central Angle and Inscribed Angle Theorem:

   • The central angle ($\angle O$) subtended by an arc is twice the inscribed angle subtended by the same arc.

3. Application: Using the theorem: $x = 2 \times 40^\circ$

4. Calculation: $x = 80^\circ$

Final Answer:

$x = 80^\circ$

Would you like a step-by-step derivation or further explanations? Here are some related questions for deeper understanding:

1. What is the relationship between the central angle and the inscribed angle in a circle?
2. How do the properties of isosceles triangles apply to this type of problem?
3. Can this problem be solved using any other circle theorem?
4. What happens to the angle if the inscribed angle is increased?
5. How do other points on the circle affect the central angle?

Tip: In circle geometry, always look for relationships between central angles, inscribed angles, and arcs. This simplifies problem-solving!