Polygons Definition and 48 Threads

In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain (or polygonal circuit). The bounded plane region, the bounding circuit, or the two together, may be called a polygon.
The segments of a polygonal circuit are called its edges or sides. The points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon.
A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons.
A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes.

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  1. L

    I Calculating the diameter of regular polygons

    I measured the maximum diameters of 15 regular polygons with side lengths of one. Is there a way to calculate the maximum diameter for regular polygons of any number of sides without using pi or any nonterminating numbers? 03-Sided = 1 04-Sided = 1.4142 05-Sided = 1.6180 06-Sided = 2 07-Sided =...
  2. Mr Davis 97

    Sum of sides of n polygons in quadrilateral is no more than 4n

    I can construct examples that are less than or equal to ##4n## quite easily, but for the life of me I cannot come with example where it's greater than
  3. S

    B Is Figure a Polygon? Examining Edges of the Image

    Can someone please tell is this (https://ibb.co/stGFSKs) figure a polygon. If yes then is the middle line would count as an edge?
  4. C

    MHB Find Probability of two polygons

    There are five hexagons. The edges of each hexagon have been colored with one of three colors randomly. If you pick two hexagons randomly without replacement, what is the probability that they are the same? (Rotation is okay). The total space or denominator is 3^(2×6), therefore we have...
  5. dave202

    B Has anyone seen this logarithmic spiral creation before?

    <Moderator's note: Image added because otherwise the thread might once become unreadable.> I have reason to believe this could have applications in physics, but right now it's just a mathematical result I came across recently. Either way, I think it is very interesting and fun to look at. This...
  6. YouWayne

    I Similar Polygon Comparison for School Project

    I'm working on a school project and my goal is to recognize objects. I started with taking pictures, applying various filters and doing boundary tracing. Fourier descriptors are to high for me, so I started approximating polygons from my List of Points. Now I have to match those polygons, which...
  7. GEOPHILE2

    A Handlebody and Knot composed of polygons

    Has anyone seen any literature related to the construction of topological structures with geometric composition as seen below?
  8. dkotschessaa

    A Fundamental and Homology groups of Polygons

    This is an old qual question, and I want to see if I have it right. I had virtually no instruction in homology despite this being about 1/4 of our qualifying exam, so I am feeling a bit stupid and frustrated. Anyway, I am given a space defined by three polygons with directed edges as...
  9. H

    I General formula for centre of mass of polygons

    The position vector of the center of mass of a triangle is ##\frac{1}{3}(\mathbf{a}+\mathbf{b}+\mathbf{c})##. Is the position vector of the center of mass of a planar four-sided figure ABCD ##\frac{1}{4}(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d})##? Does this generalise to n-sided figure...
  10. P

    I Any two polygons can be continuously "extended" or ....?

    I am looking for theorems/information related to the following statement: any polygon can be created by an infinite number of infinitely small "extensions" or "croppings" of any other polygon, such that the shape is always a polygon (after any amount of extensions of croppings). For example, I...
  11. T

    I Tiling Polygons: Can Any n-Sided Polygon Tile?

    Here is an interesting article... http://discovermagazine.com/2016/janfeb/55-pentagon-puzzler This raises the question...can any polygon with n sides be manipulated so that it will tile with other similar polygons? Can one find a shape of a 20 sided polygon that will tile with the same shaped...
  12. H

    MHB Ratio of the Area of Similar Polygons

    Two corresponding sides of two similar polygons have lengths 3 and 7. the perimeter of the larger polygon is 91 cm. What is the perimeter of the smaller polygon? What is the ratio of their areas? I believe I have found the perimeter of the smaller polygon (39), but I can't figure out the areas...
  13. S

    Face Probabilities of Archimedean Solids

    I'm looking for a way to calculate the probabilities of Archimedean Solids landing on a specific face if a person would roll one. Of course, not the regular polygons like cubes and dodecahedrons, but something with more than one type of face like the snub cube or truncated icosahedron. I am...
  14. B

    Adjacent vertices in convex polygons

    While reading a bit about dihedral groups, I encountered a curiosity regarding convex polygons that I'm not sure is true or false. Given a convex polygon P, let A and B be adjacent vertices of this polygon and let C be a vertex of P not adjacent to A. Then is it necessarily the case that...
  15. ellipsis

    Relationship between constructibility of reg. polygons and cot(pi/N)

    Full title: Relationship between the constructibility of regular polygons and the reducability of trigonometric functions into expressions of square roots. I stumbled upon this after I derived the formula for the area of a triangle given it's side length x as a trigonometry exercise. ## A =...
  16. N

    MHB Find Area of Similar Polygons (in Terms of n)

    Good Day, I can't solve the following problem because I don't know how to find the length of the first polygon. That's why my expression for the total area has 2 variables instead of just n. Any help/ advice on how I can get an expression for the total area in terms of n will be greatly...
  17. D

    Bounding General Shapes with Polygons, Especially Concave

    I apologize if this is the wrong forum but I need access to mathematicians who know what's happening with polygonal math. I created an unproven algorithm (or heuristic) back in 1999/2000 for bounding shapes with polygons. It was interesting because it was fast, general for polygons of any...
  18. J

    Polygons: Areas Homework Help & Hints

    Homework Statement (I have attached the problem to this post as a file) Homework Equations In class we learned other fomulas for the area of a triangle using the SAS case, ASA and SSS (Heron's). The Attempt at a Solution I am honestly so confused with this one. I have a bunch of random...
  19. S

    Falling polygons: meshing vs. stacking - analytic solution needed

    Falling polygons: meshing vs. stacking -- analytic solution needed I'm a game developer and not a mathematics specialist, so I'm not 100% sure if this question is correctly categorized. My problem is as follows. I'm building a game that's similar to Tetris, but in 3D instead of 2D...
  20. I

    Can you triangulate a triangle? (also, odd sided polygons to represent surfaces)

    EDIT: My guess to the below question is that no you can't triangulate a triangle because a legitimate triangulation each edge can only be linked up to exactly two distinct faces, so if you just have one triangle, each edge would be linked up to one face (the face of the triangle) I'm really...
  21. L

    Realisation possibilities for polygons

    http://books.google.rs/books?id=vrcHC9XoHbsC&pg=PA249&lpg=PA249&dq=Realisation+possibilities+for+polygons+Nolting&source=bl&ots=5uRJp8dzJe&sig=ELypwAJpy-cydUN055Sos_Gdz1Y&hl=sr#v=onepage&q=Realisation%20possibilities%20for%20polygons%20Nolting&f=false \rho(m,n)\leq 3^{2m+2n} And this...
  22. S

    Solving Geometric Puzzles: Prisms & Balloons, Polygons & Tension

    Hi everyone, I've been thinking about these questions and would like to know if you can come up with their answers: 1. if you attach prisms to a balloon and inflate it, will the prisms make it impossible to keep inflating it at some time? Talking about "formulas", why? 2. why do grounds...
  23. C

    Proving Area of n-sided Polygons Maximizes When they are Regular?

    Hello, I'm a math tutor at a community college, and one of the students recently asked me why it is always true that a *regular* polygon (regular meaning equiangular and equilateral) has maximum area for any given perimeter. It makes perfect intuitive sense, but neither I nor any of the other...
  24. J

    Irregular polygons - Equal area under horizontal line

    I've been trying to find an algorithm for finding a line passing through any 2-dimensional polygon that will divide the shape horizontally or vertically into two sections that each have the exact same area, or rather exactly half the area of the whole. I've prepared some simple examples below...
  25. P

    Polygons can be designated as N-gons. What about vertices?

    Hi In 3D modeling world, people refer to 3 sided polygons as tris, 4 sided as quads and >4 sides polygons as N-gons. Now, if I understand right, actually N-Gon refers to any. So instead of saying Triangle or Quadrilateral, I can say 3-gon and 4-gon and it would be correct, right? What about...
  26. Dembadon

    Defining Polygons with Precision: A Review of Basic Polygon Terminology

    I'm working through the following book: Principles of Mathematics, by Allendoerfer & Oakley. Since I haven't taken a proof-based course yet, and won't be able to until spring 2012 , I want to make sure that I'm not forming habits that will hurt me when I do. There are some answers that aren't...
  27. O

    Radius of Gyration for Generic Polygons

    I need to calculate the radius of gyration for a generic, convex polygon, where the density is constant, the axis of rotation is the centroid (which is known), and the positions of the vertices are known. Does such an equation exist?
  28. E

    Algebraic expression for triangles within polygons

    Homework Statement If a polygon has n>=4 sides, what is the probability, in terms of n, that a triangle made up of vertices of the polygon shares at least one side with the polygon. Homework Equations The Attempt at a Solution Treating vertices of polygons as possible outcomes...
  29. O

    Geometry with polygons formed within drawn stars

    Homework Statement I'm working with the polygons on the interior of stars that have been drawn in a specific manner. For an example, I'm currently using a 16-gram. To construct the same one I have, you array sixteen vertices equidistant from one another around a central point (as in the...
  30. J

    Geometry of circles and polygons.

    I have found an equation which deals with regular polygons touching circles tangentially with each of their sides. P=Dn\tan(\frac{180}{n}) where P is the perimeter of the polygon. D is the diameter of the circle. n is the number of sides on the polygon. i originaly thought it would be...
  31. S

    Gravitational Motion of Masses on Polygons: n=8 to ∞

    Suppose a n-sided polygon. Point particles of mass "m" each are placed in the corners of the polygon. How does the system of particles move if the only force anting between them is gravity? After how much time the bodies collide if n= 8 and n tends to infinity? Any suggestions are welcome...
  32. J

    Area of circle using incribed polygons

    I was trying to find the area of a circle the ancient way. For example, here is the area of an octagon inscribed in a circle. You formula is the same regardless of how many sides your figure has: (S/2)r2sin(2π/S). And so, the area of a circle must be limS-->∞(S/2)r2sin(2π/S). I can expand...
  33. B

    No Quadrilateral/Pentagon Knots: Simple Closed Polygons are Trivial

    Hi, everyone: I would appreciate any help with the following: I am trying to refresh my knot theory--it's been a while. I am trying to answer the following: 1) Every simple polygonal knot P in R^2 is trivial.: I have tried to actually construct a...
  34. D

    Finding # of Sides in Polygon Given Measure of Interior Angle

    Homework Statement the measure of an interior angle of a regular polygon is given. need to find the number of sides in the polygon. i cannot find the formula to be able to do this. Homework Equations The Attempt at a Solution
  35. R

    Distance between polygons in 3d? (points of contact as a function of time)

    I was just wondering, is there a way to write a polygon as an equation in 3d? (Yes, a polygon, NOT a polyhedra) It's intended to be a part of a collision detection program, so I need to be able to represent all points on a given polygon as an equation. Each polygon is being acted upon by...
  36. N

    How Does a Java Program Determine Collision Points Between Rotating Polygons?

    Okay, I am struggling severely and need some guidance by anyone who understands what's going on with this. I have to write a Java program that will take two different polygons with an angle and a pivot point. The program will determine whether the second polygon hits the first polygon as it...
  37. J

    The two dimensional regular polygons.

    The two dimensional regular polygon series, the triangle, square, square, pentagon etc. is infinite. If for some reason, it was finite, what would our universe become? especially the dimensions of length and breadth.
  38. M

    Calculating Pi with Polygons: Is Converting to Degrees a Valid Method?

    i think found a formula to calculate pic (almost). Problem is, it has pi in it. If you are working in radians, it is Ntan(π/N), where N is the number of sides of a polygon that is close to a circle. as n approaches infinity, the function approaches π. Is it cheating if i simply change it to...
  39. quasar987

    Fundamental polygons and surfaces

    The theorem of classification of closed surfaces says that any closed surface is homeomorphic to a fundamental polygon in the plane. I was wondering if any fundamental polygon can be made into a closed surface by adjoining an appropriate atlas to it. The topological requirements of a closed...
  40. Loren Booda

    Series of superimposed regular polygons

    Superimpose concentric regular polygons of equal area with maximal symmetry, starting with the equilateral triangle and sequentually approaching the circumference of a circle. What series can you derive for the fraction of the area not occupied by any successive polygons?
  41. MathematicalPhysicist

    Cardinality of concave polygons' set.

    i need to find the cardinality of the set of all concave polygons. i know that each n-polygon is characterized by its n sides, and n angles, but i didn't find its cardinality, for example we can divide this set to disjoint sets of: triangles,quandrangulars, etc. we can characterize the...
  42. S

    3D shapes with perfect polygons

    Hello, (edit should be regular polygons in title) I have been thinking a lot recently about 3D shapes formed by 2D regular polygons. I was asking myself if there would be any way to calculate the minimium number of regular polygons to form a complete 3D shape. It is fairly easy with an...
  43. K

    Fortran Drawing Polygons in Fortran: How to Use Code to Create Geometric Shapes

    Anyone know the codes to draw polygons in fortran. Actually a few points work too.
  44. W

    Constructing 3D Figures from Polygons

    By 3d figure I mean what a pyramid with a triangular base is to a triangle and what a cube is to a square. My question is, is it possible to form a tridemensional figure with any polygon? If so, what is the relation between the number of sides of the polygon and the number of faces of the figure?
  45. S

    Area, Perimeter & Radius of Circles & Polygons

    This has been bugging me for a while and I thought that you guys might know an answer. Awhile ago I realized that there is a direct relationship between the radius (as in the distance between a corner and the center) squared and the area of any regular polygon with the same number of sides. For...
  46. R

    What is the measure of each interior angle for a regular triangle?

    The questions asks me to "Find the measure of each interior angle for a regular triangle. Then Round to the nearest tenth if necessary" So, the solution of the measures of the interior angles of a triangle is 180 degrees. There's 3 angles. So each angle = 60. So all i really want to...
  47. S

    Any Facts about Newton Polygons

    I'm giving a talk in a few days about Newton Polygons for polynomials and I was wondering if anyone knew of a few (short) interesting uses I could discuss or perhaps a text with a chapter on the subject. I have a cool result about irreducibility and of course the basics but it would be nice to...
  48. J

    Centroid of irregular polygons

    Hi there, I have a bit of a problem for you. I have recently had to write a program to compute the centroid (centre of area) of a 2d shape. I used a many-point weighted triangle method. The shapes themselves are ROI's of anatomical features on SPECT and MRI scans. Im writing up my...
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