# Questions tagged [polygons]

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

803 questions
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### Find polygon that traverses all given points with minimal circumference but has a single surface

I know about Convex Hull, but convex hull is convex, which isn't what I want. I want a polygon that will always try to minimize it's perimeter however will not have multiple surfaces: For example, I ...
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### Angles between vectors of complex polygons

currently I try to get an insight into the following problem. Consider a (closed) complex polygon (edges can intersect) consisting of $n$ vectors. For simplicity, assume that every vector has unit ...
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### The relationship between the median,sides and the height of a triangle

Let $BC = a$, $b = CA$, $c = AB$ be side lengths in triangle $ABC$. Indicate the $h$ length of the height against the side $AB$ and the $m$ the length of the median from the corner $C$. Below are four ...
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### Complex coordinates of the vertices of a polygon

If $z_0$ be the centre of a regular polygon of $n$ sides and $z$ be its one vertex $A_1$, then the vertices $A_2, A_3,\dots, A_n$ (proceeding in anticlockwise direction, taking actual position of ...
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### Finding the centre of the largest inscribed circle in an irregular polygon

[edited title] I need to find A centre point of an irregular polygon. Being an irregular polygon, I'm aware that there can be no geometric centre. I have a very specific shape in mind, it is a ...
1answer
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### Determine if a point lies in a quadrangle [duplicate]

I have a quadrangle which sides consist of parts of rays, and I only know the coordinates of two points on each ray. I need to determine if a point $(x,y)$ lies in such quadrangle. In this picture, ...
1answer
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### Number of ways to choose a closed path of given length on a square lattice

Also known as self-avoiding polygons, this is an unsolved problem. However, to leading order in the asymptotic limit, the number of polygons of a given perimeter scales exponentially with perimeter ...
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### 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 ...
2answers
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### 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?
1answer
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### 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-...
2answers
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### 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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### 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 ...
1answer
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### 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 ...
1answer
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### 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!
1answer
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### 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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### 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 ...
1answer
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### 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 ...
3answers
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### 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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### 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 ...
1answer
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### 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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### 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 ...
2answers
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### 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 ...
1answer
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### 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 ...
1answer
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### 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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### 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 ...
2answers
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### 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 ...
1answer
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### 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 ...