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Questions tagged [polygons]

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

3
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What is the length of $x$ in this pentagon diagram?

ABCDE is a regular pentagon. $\angle AFD = \angle EKC$ $|FH|=1$ cm; $|AH|=3$ cm What is $|DK|?$ I know that triangles $EFA$ and $DEK$ are similar and that $|EK|=4$ cm. Also because this is ...
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0answers
27 views

how to find if it is possible to form a convex polygon with given n sides length? [duplicate]

I have given n lengths. I have to decide whether it is possible to construct an n-sided polygon with those n lengths. What are the necessary conditions or formula to check it?
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0answers
17 views

Rotate and Translate Irregular Polygon to X-axis

I need to rotate and translate an irregular polygon so that a chosen edge is on the x-axis and the "inside" of the polygon is above the x-axis. I know how to translate and rotate a polygon using a ...
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0answers
19 views

Finding the vertices of a regular n-sided polygon using its centroid

How do you find the vertices of a regular $n$-sided polygon using its centroid, with the knowledge that each vertice is distance $d$ from the centroid.
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1answer
24 views

Calculate Area of 4 points polygon from distance between poins

I have 4 points on ground, something like what is depicted in this sample: https://www.mathopenref.com/coordpolygonarea.html The problem is that I don't know how to determine the coordinates of my ...
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2answers
45 views

Area of a quatrefoil inside a circle

What is the maximum area of a quatrefoil that is inscribed in a circle of radius 6? My first guess was to cut the circle into small regions but that doesn't seem to work. The solution does not have ...
0
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1answer
24 views

Polygons with two different angles [closed]

I have the following questions: Let us consider polygons with only two different interior angles $\alpha$ and $\beta = \pi - \alpha$. For which frequencies of these angles is it possible to form a ...
4
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1answer
44 views

Convex polygon inscribed in a circle with vertices that create matrix. Prove that the rank of this matrix is less or equal 2.

Consider a convex polygon inscribed in a circle with vertices $P_1$, ..., $P_n, \ n \ge 3$. Let $A$ be the matrix $n \times n$ such that $\begin{equation} a_{ij} = \begin{cases} |P_iP_j| ...
2
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2answers
77 views

Circle measurement of Archimedes

Let $f_n$ or $F_n$ be areas of the regular $n-$ polygon described to the unit circle or circumscribed. Show $f_{2n}=\sqrt{f_nF_n}$ and $F_{2n}=\frac{2f_{2n}F_{n}}{f_{2n}+F_n}$ In the solutions ...
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2answers
58 views

What is the length of the side of a regular hexagon and why?

I think it is 1. Because from the Picture it Looks like that but can somebody explain me why this is so?
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1answer
49 views

What is the symmetry point group of a regular n-gon, where n is even with the opposite vertices identified? Is it $D_n$? [closed]

If we have a regular 2n-gon, what is its symmetry group if we identify the opposite vertices (the ones that are on the same line). Do we get the group that is isomorphous to the group of the regular n-...
3
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2answers
53 views

How to find lengths and corner coordinates of an irregular pentagon

How do I find the side lengths and therefore corner coordinates of a pentagon with the following internal angles: A = 140°, B = 60°, C = 160°, D = 80°, E = 100° ?...
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0answers
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FIGURATE / POLYGONAL NUMBER QUESTION

I recently saw the posts on 'reversing figurate number' evaluations at these links on the forum here ; http://www.scienceforums.com/topic/21846-figurate-numbers-deriving-backwards/ http://www....
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0answers
12 views

How to compare centrality of points in different shaped and sized polygons

Suppose I have several polygons of varying shape and size, each containing a single point. I want to compare the centrality of each point, and determine which point is more central to its polygon. So ...
4
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1answer
38 views

How can I find the measure of every angle in a star polygon?

I'm unfamiliar with these kinds of problems. I looked up some formulas and it says for an $(n,3)$ family of star polygons, $\theta = \frac{(1-\frac{6}{n}}{180}$ How do I get these formulas and what ...
1
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1answer
26 views

Side length of a decagon given the circumradius, without trigonometry

How can I find the side length of a regular decagon, given a circumradius R? I know how to do it with trigonometry, is there another way using Ptolemy's Theorem? Thanks!
0
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1answer
44 views

Determine Direction of Normal vector of Convex Polyhedron in 3D

How can I determine direction(point inside or outside) of normal vector drawn on one side of the polyhedron? Known informations; coordinates of all corners in 3d as x,y,z which face of the normal ...
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0answers
11 views

How to create a convex polygon which is the superset of a given polygon?

I have a JTS_footprint like this, POLYGON ((46.542770 -65.500984,54.130459 -63.660416,59.213402 -66.664993,51.004562 -68.736069,46.542770 -65.500984)) How do I ...
1
vote
1answer
43 views

Squares on the sides of a regular pentagon

On each side of a regular pentagon a square lying outside the pentagon is constructed. (see the picture below) $X_1,..., X_5$ are the centers of the squares. $P$ and $Q$ are points of intersection ...
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3answers
37 views

How would I find the lengths of the sides of a regular pentagram if given the perimeter of the circle that the pentagram is inscribed to?

For a Turtle program in Python that I am working on, I will need to draw out a star. This star is a regular pentagram, meaning that each of the sides are of equal length. As well, the pentagram is ...
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0answers
14 views

Intersection of isometries of a polygon

Suppose we have a two dimensional polygon $P$. And a sequence of polygons $(P_i)$, where each $P_{i+1}$ is a small translation/rotation of $P_i$. I am interested in situations where $\cap P_i$ is ...
1
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1answer
29 views

How to measure an angle in a polygon that is more than 180?

Assume we have an arbitrary polygon that has no holes nor self edge intersections, but can otherwise be concave and deformed. Assume the vertices are ordered either clockwise or ccw. So for example ...
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0answers
31 views

How to pick a “center” of a concave polygon?

I asked a question on how to scale concave polygons and a couple of people suggested some very clever solutions. The issue is that these solutions rely on picking an appropriate point $C$ in the ...
2
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2answers
45 views

How to scale a polygon such that all the points lie within the original polygon?

With a convex shape, like a circle, we can create a set of similar shapes, all contained within one another, by centering the shape at the origin and scaling it. So we can get the following: With a ...
2
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1answer
66 views

Centroid within non-convex 2d polygon

The centroid of an object is defined as the arithmetic mean of all points of the object. For non-convex objects, the centroid is often not a part of the object itself: Is there a definition of a ...
0
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1answer
28 views

Algorithm for converting a coordinate into angles of a pentagon.

I will go ahead and admit, this might just be something obvious but I did research and couldn't find anything. I have a pentagon, and I know two top points (A & B) and the distance between them (...
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0answers
22 views

The best approximation method to recover original polygon outline before rasterization procedure

I have a polygon, originally created as a Bézier Curve (black outline on the picture), and then saved as a polygon with enough points to call it smooth (at this scale). Then this polygon was ...
21
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2answers
2k views

If a regular polygon has a fixed edge length, can I know how many edges it has by knowing the length from corner to its center?

So I wonder if there is a formula so that when there's a defined edge length, I can calculate a regular polygon's edges amount by knowing its length from corner to center, or vice versa. So let's ...
0
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1answer
31 views

What is a non-concave and non-convex polygon called?

I am writing a software function to plot the outer points of an n-sided polygon and I'm having trouble ensuring I use the correct terminology. The function I've written simply renders the calculated ...
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0answers
72 views

Determining the value of $\frac{BC}{CE}$ in a cyclic pentagon $ABCDE$.

Let $ABCDE$ be a cyclic pentagon, where $AC=2, AD=3, BD=5, BE=1, \frac{CD}{DE} = \frac{10}{3}$. What is the value of $\frac{BC}{CE}$? I worked with the area of specific triangle with trigonometry. ...
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0answers
58 views

Cut a convex polygon in two equal areas with minimum perimeter

Given a convex polygon. How to find a cut that divides the polygon in two equal area parts and the length of this cut is minimum. Possible solution is 1. Find a minimum polygon projection. 2. Have ...
0
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3answers
47 views

Single-Line Equation for Equilateral Triangle

Is it possible to come up with a single-line equation in rectangular coordinates for an equilateral triangle with circumradius $R$, positioned symmetrical about the $y$-axis, as shown in the diagram ...
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0answers
47 views

What does the determinant of 3x3 matrix mean? how does the determinant tell me the orientation of polygon?

I need help with understanding the problem. The coding isn't really the problem rather the math part. I don't understand what a delta test method that tells me determinant of a 3x3 matrix is supposed ...
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0answers
38 views

Solid angle created from irregular polygon (over a sphere)

I have an $n$-polygon on a sphere ($n\geqslant3$). In this example the vertices are $C,D,E,F,G,H,I,J,K$. Which solid angle alpha generate this polygon respect origin of the sphere? For $C,D,E,F,G,H,I,...
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1answer
44 views

Chord partition of regular polygon: same fraction of area and perimeter?

This is a variation of a question posed by James Tanton on Twitter. Let $P$ be a regular $n$-gon, $n \ge 3$. A chord $c$ of $P$ is a segment connecting two distinct points of the boundary of $P$, on ...
1
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1answer
67 views

Why is the sum of all external angles in a convex polygon $360^\circ$ and not $720^\circ$?

Why is the sum of all external angles in a convex polygon $360^\circ$? From my understanding, for each vertex in a convex polygon, there exist exactly $2$ exterior angles corresponding to it, which ...
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3answers
48 views

Cyclic pentagon $ABCDE$ has radius three. If $AB=BC=2$ and $CD=DE=4$, find AE. [closed]

Given cyclic pentagon $ABCDE$ with the radius $3$. If $AB=BC=2$ and $CD=DE=4$, find $AE$. I think trigonometry tricks are really useful on this problem, but I still can't get the final answer.
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1answer
29 views

Maximum surface area of polygons with sequential side lengths. [closed]

What is the maximum area of a polygon with sides of lengths $1,2,3,\ldots,N$? Intuition tells me the polygon must be inscribed in a regular polygon with $1+2+3+ \cdots +N$ sides. What would be the ...
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1answer
65 views

Algorithm to construct irregular polygon

I have number of line segments (they represent walls in floor scheme) each accompanied with length and adjacent angle. What sequence of steps should my algorithm perform in order to obtain set of ...
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0answers
8 views

Non-classical examples for generalized quadrangles

I have been told that there is no non-classical example for n={6,8} known yet for quadrangles. Could you share some study on it?
0
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1answer
58 views

general equation for a n-side regular polygon

I was revisiting some geometry problems, and i got me thinking if there is any king of general equation to describe a n-side polygon? Some way similar to the equation that describe a circle, which we ...
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0answers
35 views

Polygon with all sides with different lengths

I searched a lot but I could not figure out the property name of a polygon with all sides with different lengths. First of all I am assuming that it is possible to have at least one configuration for ...
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1answer
31 views

Numerical Analysis - $n$-sided polygon tangential

i need help with this question..I'm not so sure how to go about the arguments. Any help would be appreciated. Consider a regular $n$-sided polygon tangential to and enclosing the unit circle to ...
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0answers
25 views

Polygons defined by lengths of sides

Can a polygon be defined by the lengths of its sides? In other words, if given the lengths of the sides of a polygon, is there a way to figure out what polygon has those lengths and show that there ...
0
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1answer
32 views

Which of two vertices has the wider angle?

Given that: two vertices from different paths share the same point; the winding orientation of these paths is unspecified; the angle at both vertices will be less than 180 degrees these paths don'...
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2answers
30 views

Plotting points on a halfcircle, given diameter and facing direction.

I know the coordinates of point $1$ and $2$ and some radius $r$ at a halfcircle with centerpoint point $1$, with the gap of the halfcircle pointing towards point $2$. How do I compute the (lets say $...
0
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1answer
161 views

Check if latitude & longitude coordinates are inside a specific range of a latitude longitude polygon

Since I'm not a mathematician I came here to ask what the most efficient way is to check if latitude and longitude coordinates are inside a range (for example 50 meters) of multiple latitude and ...
8
votes
1answer
66 views

Internal angles in regular 18-gon

This (seemingly simple) problem is driving me nuts. Find angle $\alpha$ shown in the following regular 18-gon. It was easy to find the angle between pink diagonals ($60^\circ$). And I was able to ...
2
votes
1answer
52 views

What is the length of the hypotenuse?

We have $n$ isosceles-right-angled triangles. The hypotenuse of the $n^{\textrm{th}}$ triangle is the base of the $(n+1)^{\textrm{th}}$ triangle. For the first triangle, $T_{1}$, the length of the ...
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1answer
19 views

Calculating square meter area with polygonal geographical coordinates (metric - not DMS system)

I'm working on a program but my problem is not on software side but mathematical. I have the following input : ...