To find the area of the circle circumscribed around the octagon, we need to calculate the radius of the circle using the given information. Let’s solve step by step.

Step 1: Understanding the problem

A regular octagon is inscribed in a circle. The given information is that the distance between two parallel sides of the octagon is $d = 5 + 5\sqrt{2}$.

This distance is equal to twice the apothem ($a$) of the octagon because the apothem is the perpendicular distance from the center to one of the sides. Hence: $2a = 5 + 5\sqrt{2} \quad \implies \quad a = \frac{5 + 5\sqrt{2}}{2}.$

Step 2: Relationship between the apothem and radius

The apothem of a regular octagon is related to the radius $r$ of the circumscribed circle by the formula: $a = r \cos\left(\frac{\pi}{8}\right),$ where $\frac{\pi}{8}$ is the angle corresponding to 22.5° (half of the angle of a central triangle in the octagon).

Rearranging for $r$: $r = \frac{a}{\cos\left(\frac{\pi}{8}\right)}.$

Step 3: Substitute $a$ into the formula

Substitute $a = \frac{5 + 5\sqrt{2}}{2}$: $r = \frac{\frac{5 + 5\sqrt{2}}{2}}{\cos\left(\frac{\pi}{8}\right)} = \frac{5 + 5\sqrt{2}}{2\cos\left(\frac{\pi}{8}\right)}.$

Using the trigonometric value $\cos\left(\frac{\pi}{8}\right) = \sqrt{\frac{2 + \sqrt{2}}{2}}$: $r = \frac{5 + 5\sqrt{2}}{2 \cdot \sqrt{\frac{2 + \sqrt{2}}{2}}} = \frac{5 + 5\sqrt{2}}{\sqrt{2 + \sqrt{2}}}.$

Simplify by rationalizing the denominator: $r = \frac{(5 + 5\sqrt{2})\sqrt{2 - \sqrt{2}}}{\sqrt{4 - 2}} = (5 + 5\sqrt{2})\sqrt{2 - \sqrt{2}}.$

Step 4: Area of the circle

The area of the circle is: $\text{Area} = \pi r^2.$

Substitute $r = (5 + 5\sqrt{2})\sqrt{2 - \sqrt{2}}$: $r^2 = \left((5 + 5\sqrt{2})\sqrt{2 - \sqrt{2}}\right)^2 = (5 + 5\sqrt{2})^2 (2 - \sqrt{2}).$

Expand $(5 + 5\sqrt{2})^2 = 25 + 50\sqrt{2} + 50 = 75 + 50\sqrt{2}$: $r^2 = (75 + 50\sqrt{2})(2 - \sqrt{2}).$

Distribute: $r^2 = 150 - 75\sqrt{2} + 100\sqrt{2} - 50\sqrt{4} = 150 + 25\sqrt{2} - 100 = 50 + 25\sqrt{2}.$

Thus, the area of the circle is: $\text{Area} = \pi r^2 = \pi (50 + 25\sqrt{2}).$

Final Answer:

The area of the circumscribed circle is: $\boxed{\pi (50 + 25\sqrt{2})}.$

---

Would you like further clarification or assistance with any step?
Here are some related questions to explore:

1. What is the relationship between the radius of a circumscribed circle and the side length of a regular polygon?
2. How do you derive the formula for the apothem of a regular polygon?
3. Why does the angle $\frac{\pi}{8}$ appear in the calculation for the octagon?
4. Can you calculate the area of the octagon given the same apothem?
5. How does the area of the circle compare to the area of the octagon?

Tip: Familiarize yourself with common trigonometric values like $\cos\left(\frac{\pi}{8}\right)$ and $\sin\left(\frac{\pi}{8}\right)$ for polygons inscribed in circles!