For questions on polygons, a flat shape consisting of straight lines that are joined to form a closed chain or circuit

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When is area of regular $n$-gon expressible by radicals?

I had a look at the table here. As there are only $20$ values published I noted that the area values given as radicals ( of square roots ) correspond to values of $n$ for which the $n$-gon is ...
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1answer
32 views

With $r_{n-1} = \frac{r_{n-2}}{\cos(\frac{180}{n})}$, does $\sum_{n=3}^\infty r_{n-1}-r_{n-2}$ converge?

Let $r_{n-2}$ be the the inradius and $r_{n-1}$ the circumradius of a regular n-gon. From this (1) and this (14) we get: $$r_{n-1} = \frac{r_{n-2}}{\cos(\frac{180}{n})}$$ Basically what I'm doing ...
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48 views

Is a 2-sided polygon in 2D a reasonable concept, and what is then the definition?

A 5-sided polygon can be reduced to a 4-sided polygon by removing one vertex and the two connecting edges, and re-connecting the two adjacent vertices with an edge, whereby the interior angle goes ...
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41 views

Solving the 2D Wave Equation on a Polygon Membrane

Is it possible to solve the 2d wave equation on a Polygon membrane (such as a hexagon)? And if so, would separation of variables hold in this case?
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36 views

Given angles/sides, is there a necessary and sufficient condition on whether polygon exists?

We are given the angles and side lengths of a polygon $a,b,...$ and $A,B,...$ like so: (the polygon has non-negative side lengths and is convex) How can we find out whether a polygon with these ...
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1answer
142 views

Overlapping area between a circle and a square

I have a circle and a square. They are aligned to their center. The radius of the given circle is less then half the value of diameter of the square. How to find the overlapped area?
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1answer
1k views

How many triangles can be formed by the vertices of a regular polygon of n sides?

How many triangles can be formed by the vertices of a regular polygon of $n$ sides? And how many if no side of the polygon is to be a side of any triangle? Got no idea where I should start to think. ...
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23 views

Theorem references

Yesterday night i think i proved the following theorem: theorem among all the possible regular polygons inscribed in a given circle, the hexagon is the only one with a side length equal to the ...
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1answer
85 views

Relationships between Schläfli symbol and geometrical properties of regular concave (star) polygons

In a Geometry class, we have been learning about the Schläfli symbol and how it is used to describe regular polygons with the notation {p, q}. I am studying a special class of concave polygons known ...
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1answer
50 views

What does a circle with a dot in it represent?

Consider the following formula for finding the area of an arbitrary planar polygon: $$ A=N\odot\sum_{i=0}^{n-1}\left(\frac{P_i\times P_{i-1}}{2}\right)$$ Where N = the plane's normal, and Pi ...
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2answers
33 views

Describing the symmetries of a $2n$-gon in $\Bbb R^2$ with matrices.

Problem: Consider a regular $2n$-gon in the Euclidean plane $\Bbb R^2$ centered at the origin $(0, 0)$ and with its $2n$ vertices equally distributed on the unit circle. Label the vertices from ...
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0answers
43 views

All polygons satisfy the “normal” property.

A fancy explanation is below, but here's an edited simpler explanation because I think the jargon makes the problem seem inaccessible. In reality this problem is super accessible and I'm sure the ...
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1answer
42 views

Proof of Jordan Curve Theorem for Polygons

So I'm trying to prove the Jordan curve theorem for polygons, but I'm not sure how to show that a line segment that does not intersect the boundary has all points of the same parity. I'm also not ...
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2answers
81 views

Polygons with a Unique Triangulation

For each n > 3, find a polygon with n vertices that has a unique triangulation. I want to say that you can somehow build these polygons by continuously adding triangles somehow, but I'm not sure.
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4answers
179 views

Fastest method to draw constructible regular polygons

We know from Gauss, that the regular polygons of order $3$, $4$, $5$, $6$, $8$, $10$, $12$, $15$, $16$, $17$, $20$, $24\ldots$ are constructible. Is there a provably fastest compass and straightedge ...
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1answer
118 views

Cartesian coordinates for vertices of a regular polygon?

I'm trying to draw: A set of $N$ (edit) irregular polygons one inside the other, where the innermost should be an equilateral triangle, enclosed by a square, enclosed by a pentagon, etc. Where ...
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1answer
122 views

Sum of the vectors from one fixed vertex to each remaining vertex of a regular polygon

I'm attempting to calculate the sum of the vectors from one fixed vertex of a regular m-sided polygon to each of the other vertices. It's for a study guide preceding my Linear Algebra exam tomorrow, ...
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0answers
25 views

decomposition of Non-convex polygon

Is it possible to decompose a non-convex polygon, with more than one of its interior angles greater than 180, into a number of convex polygons ? If so, how is it possible ? Is there any algorithm for ...
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3answers
50 views

$\cos \alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+…+\cos(\alpha+(n-1)\beta)=0 $

If each side of a regular polygon of $n$ sides subtend an angle $\alpha$ at the center of the polygon and each exterior angle of the polygon is $\beta$,then prove that $\cos ...
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2answers
58 views

If $a_1,a_2,a_3,…,a_n$ are the side lengths of $A_1A_2A_3…A_n$ convex polygon,then$\frac{a^2_1+a^2_2+a^2_3+…+a^2_{n-1}}{a^2_n}$ is

If $a_1,a_2,a_3,...,a_n$ are the side lengths of $A_1A_2A_3...A_n$ convex polygon,then$\frac{a^2_1+a^2_2+a^2_3+....+a^2_{n-1}}{a^2_n}$ is ...
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0answers
54 views

What is the probability to pass through $1\le m\le n$ vertices of an $n$-sided polygon after $t$ seconds?

Suppose a flea is on a vertex of an $n$-sided polygon. It stays still for exactly one second, and then jumps instantly to an adiacent vertex. Let us assume it has no memory of its previous jumps and ...
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2answers
175 views

What are the rules for a Tetartoid pentagon?

The tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry. It has twelve identical ...
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5answers
156 views

$\frac{1}{A_1A_2}=\frac{1}{A_1A_3}+\frac{1}{A_1A_4}$.Then find the value of $n$

If $A_1A_2A_3.....A_n$ be a regular polygon and $\frac{1}{A_1A_2}=\frac{1}{A_1A_3}+\frac{1}{A_1A_4}$.Then find the value of $n$(number of vertices in the regular polygon). I know that sides of a ...
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2answers
54 views

O.I.M. polygon inequality

I am trying to prove an inequality which was used to prepare the Romanian O.I.M. team. I seem to lack ideas on how to tackle this problem. We take a convex polygon $P_1\ldots P_{n+2}$ and consider ...
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2answers
32 views

Geometrical calculation to enlarge the height of rotated rectangle

There is a polygon (rotated rectangle) that defined by 4 corner points in 2D coordinate system. Does anyone help me with the fast (minimum trigonometry operations) algorithm to change its height by ...
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2answers
85 views

Area of regular n-gon without trig?

As the title suggests I'm trying to find a formula for the area of a regular n-gon that doesn't use trigonometry. I already know the trig formula and I realize that my question is simply asking for ...
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2answers
55 views

Polygon and line intersection

Does anyone help me with the fast algorithm to determine the intersection of a polygon (rotated rectangle) and a line (definite by 2 points)? The only true/false result is needed.
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3answers
176 views

Calculate pentagon area based on lengths of all its sides

Sorry for this question. I guessed there is an online calculator to calculate the area of the pentagon if we know lengths of all its five sides. So, here are the lengths of sides of pentagon ABCDE: ...
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1answer
35 views

Covering polygons with circles of minimal radius

I have a closed polygon and I would like to fully cover it with a set of K circles of different radius such that the area covered by the circles but outside the polygon is minimal. This seems the ...
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2answers
111 views

Area of a polygon inscribed into an ellipse

I have recently found a paper describing that the percentage area error of a polygon inscribed within a circle can be calculated using the following formula. The output of the algorithm is a set ...
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1answer
46 views

Finding center of convex polygon

If I'm given vertices of a convex polygon (in the attached image, they are D,E,F,G and H) if we know that inside the polygon there exists a point (say O) for which each angle created by any two ...
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2answers
177 views

Trying to understand the limit of regular polygons: circle vs apeirogon (vs infinigon?)

In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon: "In the limit, a sequence of regular polygons with an increasing number of sides becomes ...
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2answers
153 views

Calculate area of “hand-drawn” polygon

I have a series of coordinates that represent a hand-drawn polygon. At the intersection, the lines slightly "overshoot," e.g.: ...
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1answer
123 views

inscribed circle in $n$-gon

If I'm given a circle with radius $r$ and I want to create a polygon with side $n$ (say $n=5$) which can cover the circle fully, then how to prove that a regular polygon is the solution with minimum ...
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0answers
17 views

Formula to determine the number of exterior edges in multiple tiled hexagons

I'm looking for a formula which determines the number of external (that is, non-touching) edges in multiple tiled hexagons. By observation, there are 10 external edges when 2 hexagons are adjacent. ...
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29 views

Polygons - necessity of checking for collinearity with edge incident to diagonal's vertices?

I'm reading a book on Computational Geometry ('CG in C' by Joseph O'Rourke). It is quite enlightening but there is one thing I feel like I have to ask about when it comes to triangulation of a ...
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1answer
38 views

Equivalent definitions of symmetry group of regular n-gon (dihedral group)

Let $P_n$ be a fixed regular convex $n$-gon in the plane. For a metric space $M$ we denote by $\text{Isom}(M)$ the set of distance-preserving maps $M \to M$. How can I show that $$ D_n := \left\{\, f ...
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2answers
50 views

Is circle the only Jordan curve with this property?

When I was thinking about one problem that has to do with Jordan curves the problem which I am going to describe now, arose in my mind. And here it goes. It is known that for every $n\geq3$ the ...
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0answers
78 views

Solve for Sides of a 5-Sided Irregular Polygon

I have a 5-sided irregular polygon and I know the lengths of 4 of its 5 sides and 2 of its 5 angles. Is there a way to know the length of the 5th side using this information?
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1answer
59 views

“How to sort vertices of a polygon in counter clockwise order?”: Computing Angle?

my question relates to the answer to the following question: How to sort vertices of a polygon in counter clockwise order? I don't have a strong background in linear algebra... I don't understand ...
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1answer
104 views

Nested Regular Polygons

I've created a problem that I do not know how to answer without a huge amount of effort. If you have an elegant solution to it, please share! Take an equilateral triangle of side length 1 and ...
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0answers
36 views

Testing if a number N is prime by its regular polygon's angles

Is it possible to tell if a number N is prime by looking at the angles of a regular N-sided polygon? For example, a regular triangle has 60 degree angles, is there a way to tell that the number 3 is ...
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1answer
107 views

Intersection of a convex polygon and a moving circle

I have a straight line which intersects a convex polygon in 2D plane. There exists a circle with constant radius. The center of circle is moving on this line. So at first the polygon and circle don't ...
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1answer
32 views

Sum of Distances between Points on a Regular $n$-gon

I received a question asking to determine a formula to sum the distances between all points of a regular $n$-gon inscribed in a circle of radius $1$. To solve this, I instead worked with the ...
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133 views

Shortest system of roads between 4 cities

You have $4$ cities placed on the vertices of a square of side length $1$ km. You have to come up with a system of roads such that you can reach any city from another (directly or through another ...
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How to determine the outward normal vector of a face of a hexahedron if the orientation (CW or CCW) of vertices of the face is unknown

Consider a single quadrilaterally-faced hexahedron. If given the co-ordinates of the vertices, $\mathbf{v}_i$, of a face in counter-clockwise orientation, I can compute the corresponding unit outward ...
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1answer
17 views

Adjust Angle to Add Vector for Non-Equiangular Non-Equilateral

I asked this question: Adjust Angle to Add Vector and the solution showed that for equiangular, equilateral triangles the ratio between $\theta$ and $\phi$ was $\pi + \theta = 2\phi$: But now I ...
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2answers
76 views

Can every (convex) polygon be described by a single inequality (involving absolute values)?

For example, $$ |x| + |2x + y| + |x + 2y| + |y| + |x+y| < 4 $$ describes an octagon. I'm wondering whether an equation of this form always exist for any convex polygon, and if so, whether there ...
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2answers
378 views

Construct a regular hexagon of specific height?

Is it possible to construct a hexagon of particular height, meaning distance between the faces (not vertices)? I have seen various methods of constructing a hexagon (ruler and compass only) which are ...
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1answer
43 views

Adjust Angle to Add Vector

Given: Three 2 component vector $\vec{x}$, $\vec{y}$, and $\vec{z}$ such that $\vec{x} + \vec{y} = \vec{z}$ and $\|\vec{x}\| = \|\vec{y}\|$ $\theta$ such that the angle between $\vec{x}$ and ...