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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19
votes
7answers
828 views

Number of ways to connect sets of $k$ dots in a perfect $n$-gon

Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices in a given perfect $n$-gon such that no two lines intersect at the interior of the $n$-gon and no vertex remains ...
16
votes
2answers
429 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
15
votes
3answers
537 views

Area of the Limiting Polygon

Start with an equilateral triangle with unit area. Trisect each of the sides and then cut-off the corners. In this case, we get a regular hexagon - see the picture below. Next, trisect each of the ...
14
votes
2answers
194 views

Coffee and regular polygons

To save some money, I decided to brew my own morning-fix coffee and skip buying it from the coffee shop. BTW, I drive to work and put my coffee cup in between the two front seats. While driving on the ...
12
votes
1answer
461 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
10
votes
2answers
2k views

Can a polygon be one dimensional?

When looking up the definition of polygon, Wikipedia tells me: In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing ...
10
votes
3answers
760 views

Scaling and rotating a square so that it is inscribed in the original square

I have a square with a side length of 100 cm. I then want to rotate a square clockwise by ten degrees so that it is scaled and contained inside the existing square. The image below is what I'm ...
10
votes
3answers
123 views

Closed form for the area of a convex cyclic n-gon, given the set of edge lengths

Let's say we are given a set of positive reals, and we're told that these are the edges of a convex cyclic $n$-gon, and we must compute it's area. For $n = 3$ there is the famous Heron's formula: $$...
9
votes
3answers
4k views

How is the area of a country calculated?

As countries' or states' borders are not straight lines but they are irregular in nature, I wonder how anyone can calculate the area of a country or state. When do you think the area of a country or ...
9
votes
2answers
199 views

Why does the term ${\frac{1}{n-1}} {2n-4\choose n-2}$ counts the number of possible triangulations in a polygon?

In the given picture bellow, it counts the number of different triangloations in a polygon, how do the get to this expression, why is it: $$ {2n-4\choose n-2} $$ and why do we multiply it by $${\...
9
votes
1answer
62 views

How do you calculate the smallest cycle possible for a given tile shape?

If you connect together a bunch of regular hexagons, they neatly fit together (as everyone knows), each tile having six neighbors. Making a graph of the connectivity of these tiles, you can see that ...
9
votes
1answer
317 views

Area of a Random Polygon

The following is a long description of a computation I'd like to make. You can think of the process described as a spider randomly building a web. I'd like to know how big we can expect the web to ...
9
votes
0answers
218 views

What is the shape of the convex $n$-gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $A_n$? $$A_n=\sum_{1\le{i}\...
8
votes
2answers
197 views

How to check if polylines can be untangled?

In a program I'm writing I need to be able to check whether a straight line between two points is homotopic to a polyline between them. For example in the below example the first one is equivalent to ...
8
votes
2answers
691 views

Importance of construction of polygons

Why did the ancient Greeks give so much importance to the construction of regular polygons with $n$-sides using only ruler and compass and tried to study for what $n$ was such a construction possible? ...
8
votes
1answer
221 views

What kind of polygonal surface has an interior angle > 360°?

Consider this polygon as the setting for a dynamical billiard: When it's drawn in the plane, the polygon intersects itself; it is non-simple. However, I don't want to embed the polygon in the plane!...
8
votes
1answer
88 views

can a convex polygon have only one boundary point at locally maximum distance from its centroid?

It's easy to see that given any convex polygon P and any point c in its interior, there is at least one point m on the boundary of P at locally maximum distance from c: simply choose m to be a vertex ...
8
votes
1answer
155 views

Can the $9$ point circle be generalized to $n$-gons of $n\gt3$?

All triangles have concyclic vertices and have a $9$ point circle which intersects the triangle's feet and the midpoints of its sides (as well as $3$ other significant points). Is this special for ...
7
votes
2answers
480 views

Area of an irregular polygon

I was searching for methods on how to calculate the area of a polygon and stubled across this: http://www.mathopenref.com/coordpolygonarea.html. It does work and all, yet I do not fully understand why ...
7
votes
1answer
3k 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. ...
7
votes
4answers
224 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 ...
7
votes
3answers
298 views

Transform polygons into one another?

I am aware that there must be no standard way to achieve this, but I don't know what has been done so far. I feel like I'm missing keywords to investigate further. I have any two 2D polygons $a$ and $...
7
votes
1answer
525 views

Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number

Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number, where a number $q$ is practical if and only if every integer less than or equal to $q$...
6
votes
1answer
70 views

Calculating radius of circles which are a product of Circle Intersections using Polygons

Lets say you imagine a circle with the radius $R$ and you inscribe a regular polygon with $n$ sides in it, whose side we know will then be: $$a=2R*sin(\frac{180}{n})$$ Then you draw a set of circles ...
6
votes
4answers
254 views

Pentagon Geometry

$ABCDG$ is a pentagon, such that $\overline{AB} \parallel \overline{GD}$ and $\overline{AB}=2\overline{GD}$. Also, $\overline{AG} \parallel \overline{BC}$ and $\overline{AG}=3\overline{BC}$. We ...
6
votes
0answers
89 views

Counting Regular polygons in Complete Graphs

The figure shows the correct $24-$gon, which held all the diagonals. a) Find out how we got right triangles and squares (question for arbitrary $n$)? b) How this problem can be generalized (if it is ...
5
votes
5answers
206 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 ...
5
votes
1answer
162 views

Why do rings appear in regular polygons with diagonals?

When looking at regular polygons with all the diagonals filled in, I saw that concentric rings seem to form. Why does this occur? It's not so obvious with small $n$, but for larger $n$ it becomes ...
5
votes
1answer
96 views

Maximizing area of a pentagon

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is ...
5
votes
1answer
4k views

Why does nature prefer hexagons?

The best ratio of surface to volume in three dimensional space is the ball. This can be easily observed with soap-bubbles, rain-drops and so on. They "choose" this shape naturally. Given restricted ...
5
votes
2answers
348 views

Maximum number of vertices in intersection of triangle with box

Suppose we have a triangle and a box in 3D. The intersection of the triangle with the (solid) box will be a polygon with some number of vertices (possibly zero). The vertex count will vary according ...
5
votes
3answers
116 views

What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational ...
5
votes
1answer
114 views

Finding $\sqrt{17}$ and $\sqrt{257}$ in the regular $17$-gon and $257$-gon?

(Edit: I need to revise this question with my original intent. Pls do not answer it yet. Thanks.) Given the regular $n$-gon formed by the $n$-th roots of unity. For some $n$, how do we find $\sqrt{...
5
votes
2answers
93 views

Show that diagonals intersect at common point

Given is octagon where opposite sides are equal length and parallel. Show that diagonals: $AE,DH, BF, CG$ intersects at point $S$ So I have tried to create a parallelograms $AHED$ and $BCFG$ and use ...
5
votes
1answer
2k views

Concave polygons overlapping test

I have set of $N$ concave polygons, given as list of 2D Euclidean coordinates. How to compute: a. if any of them are overlapping? b. if one arbitrarily selected polygon overlaps with any of the ...
5
votes
2answers
61 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 $(A)>\frac{1}{n-1}\hspace{1cm}(B)<\frac{1}{n-1}\hspace{...
5
votes
2answers
80 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 ...
5
votes
1answer
343 views

Polygon sine waves

So I came across this picture on Google+ and I wanted to understand further. I created an equation for the second wave, the one with the square. Here it is: $$y=\frac{\sin x}{\cos(\min(x \mod \pi/2, \...
5
votes
1answer
506 views

cyclic polygons & trigonometry

At one vertex of a pentagon inscribed in a circle of unit diameter (unit diameter, not unit radius) let the angles between adjacent diagonals be $\alpha,\beta,\gamma$, at the next, $\beta,\gamma,\...
5
votes
0answers
107 views

Polygons with coincident area and perimeter centroids

Let $P$ be a simple, planar polygon. Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the closed shape $P$. Define $c_p$ as the perimeter centroid of $P$, the center of gravity ...
4
votes
4answers
935 views

What polyhedron has 11 vertices and 17 edges

On my math test it asked me how many polygons it takes to create a polyhedron that has $11$ vertices and $17$ edges. I'd just like to see what the shape would look like and I can figure out the ...
4
votes
2answers
2k views

Algorithm of cutting a polygon into equal parts

I have a convex polygon. I need to divide it into 4 equal parts using the two slit. For example, if I have a square, I have to cut it along the diagonals. Are there some common algorithm for this ...
4
votes
2answers
39 views

Polygons joining together to make similar polygons

I was given the below question in a math competition a few weeks ago. I was bit confused about the wording of the problem and what was meant by the word "similar" in the given context. I tried ...
4
votes
2answers
162 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 ...
4
votes
1answer
53 views

What is the name for a rectangular figure of many sides?

What is a polygon where each edge is at a 90 degree or 270 degree angle to the prior edge (giving both concave and convex vertices) called? Here is one example of such a shape: ...
4
votes
2answers
49 views

Average Perimeter With n Points on the Unit Circle

A couple days ago, a friend challenged me to solve a problem: You have N vertices, each randomly placed on the edge of a unit circle. What is the formula (given N) that yields the average perimeter ...
4
votes
2answers
350 views

splitting polygon in 4 equal parts

I have a convex polygon and I want to divide into 4 equal parts using the two perpedicular splits. Like in a picture. I need s1 = s2 = s3 = s4; I need to get coordinates of point where the lines ...
4
votes
2answers
142 views

What is the ratio of the side length of a regular hepatgon to the side length of the internal heptagon?

Given a regular heptagon with side length 1, create a star heptagon by connecting every vertice. Note that removing the "points" of the star yields a similar heptagon. I want to know the side ...
4
votes
1answer
350 views

Symmetrically splitting an octagon into quadrilaterals

I'm wondering whether it is possible to split an octagon into a finite number of quadrilaterals, such that the result is symmetric from all 8 directions (sides or points). There is one condition — any ...
4
votes
0answers
72 views

Parameterizing equilateral polygons

I'm not exactly sure how to describe what I want, so if I butcher terms, please forgive me :) I want to "parameterize" the space of simple irregular equilateral polygons with n sides, or at least a ...