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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.

Dan
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    I'm hosting a dinner party for my parents, my friend's parents, my brother, and my sister. For dessert I baked a pie. How much does each person get served? :) – Attack68 May 18 '23 at 14:29
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    @Attack68 Lol, but actually if you only have a straightedge and compass, you cannot divide a pie into seven equal sectors. – Dan May 18 '23 at 15:19
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    That is "InDirectly" Specifying the number 7 , @Attack68 , which OP wants to Prohibit ! – Prem May 18 '23 at 15:39
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    “What is the smallest positive integer $n$ for which there is no question that contains no numbers except 1 whose answer is $\pi/n$?” Oh, wait... – Dan May 18 '23 at 16:17
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    And $\pi/4$ has a much simpler question: What is the measure (in radians) of the non-right angles of an isosceles right triangle? Or equivalently, what is $\arctan(1)$? – Dan May 18 '23 at 16:20
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    @Dan: Having two users here with the same name is confusing. – Dan May 19 '23 at 00:37
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    It is confusing to a "Dan" , who knows that he is not the other "Dan" .... Imagine the confusion which others have , not knowing which "Dan" is which .... not sure which @Dan will get notified here .... !! – Prem May 19 '23 at 14:10
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    @Prem: Well, if you mouse over our username links, you can see our different profile URLs (with different user IDs) at the bottom of your browser, and different reputation scores in the tooltip text. – Dan May 19 '23 at 14:30
  • That is true , @Dan , though that comes when/after we know that there are 2 users with Same Name. – Prem May 19 '23 at 14:51
  • There can't be a $\pi/n$ than cannot be attained this way. Proof: If there were, then the question "$\pi/n$ with smallest $n$ such that $\pi/n$ that cannot be attained this way" would be a valid question with that as an answer, contradiction! – Trebor May 22 '23 at 04:26

2 Answers2

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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 ?

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 ?

Answer :
$\pi/7$

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

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

With $C=6$ , this will not work :
6 WRONG
The 2 Purple Circles are not neighbours & have no Common neighbours.

With $C=6$ , this will work :
C=6

With $C=7$ , this will not work :
7 WRONG
Here , the Purple Circle is too far from the Grey Circles.

Maximum $C$ is 7. We can arrange these Circles like this :
1+6=7

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.

With $C=8$ , there is no way.
8 WRONG
Here , the Grey Circle is too far from the Purple Circles.

Prem
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    You might want to clarify the first line of "The unit circle is dissected into C smaller circles of equal size". A dissection problem typically requires a finite number of well-behaved pieces that may be rearranged to form the other. However, my read of your problem is that you just want the circles to have the same total area. – Calvin Lin May 22 '23 at 16:05
  • Oh , good Point , I have reworded that now , @CalvinLin ! – Prem May 23 '23 at 09:00
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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?

David K
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    Not to nit-pick , but [[1]] $2\pi/7$ & $3\pi/7$ are larger & can not be Constructed. [[2]] Changing to smallest will not work because $\pi/11$ & $\pi/47$ are smaller. [[3]] We could use this fact , but a little more tweaking will be necessary. – Prem May 18 '23 at 18:10
  • @Prem: “Find the largest angle with a measure of $\pi/n$, where $n$ is a positive integer, that is not constructible.” Does that work? – Dan May 18 '23 at 18:21
  • I think that tweak works !! – Prem May 18 '23 at 18:30
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    @Prem Since the total area of the circle is $\pi$, it cannot be divided into sectors that all have equal area $2\pi/7$. But Dan's formulation is essentially the same idea. – David K May 18 '23 at 20:01
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    My idea was similar, but cast it as the largest possible angle $APB$ in a regular polygon where $AB$ is a side of the polygon, and where the regular polygon cannot be constructed by compass and straightedge. – Daniel Schepler May 18 '23 at 22:20
  • @DanielSchepler ... where $P$ is another vertex of the polygon? – David K May 18 '23 at 22:42
  • @DavidK Yes, forgot to add that. – Daniel Schepler May 18 '23 at 22:45
  • When I made my Earliest Comment , it was ambiguous whether we want largest Angle (will not work) or largest Area (will work) , though that ambiguity is removed with the Answer Edit. Ok , largest Possible angle $APB$ is $2\pi/7$ , because we can Divide the Centre Point into 7 Parts, @DanielSchepler , hence we have to tweak it a little. – Prem May 19 '23 at 14:17
  • @Prem Indeed the central angle is $2\pi/7$, which is also why I asked Daniel if his point $P$ was another vertex. (It is, which means it gives the inscribed angle $\pi/7$ rather than the central angle.) – David K May 19 '23 at 17:42