Is there a question that contains no numbers except $1$, whose answer is $\pi/7$?

Question

Is there a question that contains no numbers, except possibly $1$, whose answer is $\pi/7$ ?

There are plenty of questions with no numbers except $1$, whose answer is $\pi/n$ for small integer values of $n$ other than $7$. For some reason, $\pi/7$ seems to be unattainable.

Examples:

1. Evaluate $\int_{-\infty}^\infty \frac{\sin{x}}{x}dx$. Answer: $\color{red}{\pi}$

2. Drop a needle of length $1$ onto a floor with equally spaced parallel lines a distance $1$ apart. On average, how many times do you have to drop the needle until it crosses one of the lines? Answer: $\color{red}{\pi/2}$

3. What is the volume of a cone with unit base radius and unit height? Answer: $\color{red}{\pi/3}$

4. A regular $n$-gon of side length $1$ encloses a regular $(n+1)$-gon. The polygons are concentric, and for each $n$ the area of the inside polygon is maximized. What is the limit, as $n\to\infty$, of the difference in their areas? Answer: $\color{red}{\pi/4}$

5. On a sphere of diameter $1$, uniformly random points are chosen until there is a unique circle that passes through them. What is expected area of this (planar) circle? Answer: $\color{red}{\pi/5}$

6. A regular $n$-gon of side length $1$ is inscribed in a circle. What is the limit, as $n\to\infty$, of the difference between their areas? Answer: $\color{red}{\pi/6}$

7. $\color{red}{???}$

8. Like question 4, except the polygons do not have to be concentric. Answer: $\color{red}{\pi/8}$

Indirectly specifying a number is not allowed; for example, "heptagon" is equivalent to "$7$-gon" and is thus not allowed. The question should not be obviously ad hoc. (For example: "A unit circle is divided into $k$ equal parts, where $k$ is the product of the first even prime and the first odd prime, plus $1$. What is the area of each part?") I hope the spirit of my question is understood.

Answer 1

You hinted at an answer in a comment:

A circle of radius $1$ is partitioned into sectors by radial lines, all of these sectors having equal area. What is the area of the largest such sector that cannot be constructed by compass and straightedge?

Answer 2

Question :
The Unit Circle is Dissected into $C$ Smaller Circles of Equal Size.
The $C$ Smaller Circles are arranged on the Euclidean Plane such that every Circle is Either touching every other Circle Or can reach every other Circle with a Single Circle in-between.
In other words , Every Circle is neighbours with every other Circle or it has a Common neighbour with the non-neighbouring Circles.
Every Pair of Circles will be made of neighbours or will have a Common neighbour.

What is the Minimum Area of those $C$ Smaller Circles ?

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Dissection may not be the appropriate word. I will restate with other words :
The Unit Circle is covered with White Paint. Paints in various other colours will be given to colour a Set of Smaller Circles such that the total Paint Quantity is Equal to the White Paint Quantity , which is naturally $\pi$ Units.
The Smaller Circles are required to be arranged such that every Pair $(C_n,C_m)$ have a Common neighbouring Circle or are neighbours themselves.

What is the Minimum Paint Quantity of each colour ?

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Answer :
$\pi/7$

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Elaboration :
Basically Unit Circle Area = $\pi$ , Each Smaller Circle has Area $\pi/C$.

With $C=2,3,4,5$ , we have :

With $C=6$ , this will not work :

The 2 Purple Circles are not neighbours & have no Common neighbours.

With $C=6$ , this will work :

With $C=7$ , this will not work :

Here , the Purple Circle is too far from the Grey Circles.

Maximum $C$ is 7. We can arrange these Circles like this :

Every Circle is neighbours with every other Circle Or Every Circle is neighbours with the Centre Circle which is the Common Neighbour with the non-neighbouring Circles.
We can not have larger $C$.
Hence , Minimum Area of those $C$ Smaller Circles is $\pi/C=\pi/7$.
This is the Minimum Paint Quantity of each colour.

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With $C=8$ , there is no way.

Here , the Grey Circle is too far from the Purple Circles.