What is Polygon: Definition and 114 Discussions

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. R

    I How Toppling works for a polygon shaped body already in motion, No Ex. Force applied

    How does a polygon shaped body's toppling shifts from one edge to next adjacent edge when the object is already in motion without any sliding and no external force is applied ? Explain it with the case of hexagon or octagon both ways, with and without including CoM.
  2. paulb203

    B When is a polygon not a polygon?

    For a shape to qualify as a polygon does it have to be closed. I was just introduced to frequency polygons and was surprised to see each one was a shape that wasn't closed. The definitions of polygon that I've come across go something like, a closed shape consisting of three or more lines.
  3. A

    Engineering Drawing velocity polygon for a press mechanism

    Hello, I'm new to the forum, I want to ask help on this problem here, above is a press mechanism and I'm tasked to draw a velocity polygon based on this mechanism. The point of contact between the two gears is A and consider it as a swivel hinge (rotating but stay still), so I guess it's O2...
  4. P

    I Polygon Coordinates given the Area and Center point

    I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following - Center point is known - area is known - Point A is known - Points B, C, and D are UNKNOWN I am NOT a math pro - this is for a puzzle I’m trying to solve and I can’t remember if this...
  5. A

    Finding the largest ellipse that will fit in a polygon

    Summary:: Questions about finding largest ellipse in polygon Was wondering if someone could help me understand two concepts involved with finding largest ellipse in a polygon? Some background: First, I set up a set of half-plane representations for the example polygon below and as you can...
  6. B

    Finding the resultant force using a vector polygon diagram

    Hi I made an attempt at this problem but have got the wrong answer The correct answer is actually resultant force = 21.767 N at 61.34 degrees (or 151.34 degrees bearing), but I don't know how they got that? Any help would be appreciated! Thanks
  7. E

    A polygon is rolling down a hill

    An ##n##-sided regular polygon is rolling down a frictional ramp at angle ##\theta## to the horizontal. I define ##\beta := \frac{2\pi}{n}## as the angle at the top of each of the ##n## isosceles triangles that make up the polygon. Let ##\omega_{k, 1}## be the angular velocity just after the...
  8. K

    MHB Finding 2D Polygon Coordinates from a point

    Suppose that I have the coordinates of x and y on a plane. I am writing a piece of software where the user can select a polygon of 3, 4, 5, 6 or 8 sides. All of the polygon points are equidistant from the x, y point. In other words, if you drew a circle where the center was the x, y point, all...
  9. J

    B Is this true? The area of a circle can be approximated by a polygon

    Hello everyone! I have been looking for a general equation for any regular polygon and I have arrived at this equation: $$\frac{nx^{2}}{4}tan(90-\frac{180}{n})$$ Where x is the side length and n the number of sides. So I thought to myself "if the number of sides is increased as to almost look...
  10. S

    MHB Does a line intersect a polygon

    Hello, I'm wondering if anyone has a formula for determining whether a line intersects a polygon. I would define the line with a starting latitude/longitude and ending latitude/longitude, and I would define the polygon with a series of latitude/longitude coordinates. Many thanks in advance...
  11. Benjamin_harsh

    What do questions based on the polygon law look like?

    I am searching all over google, I didn't find any problems on polygon law.
  12. M

    MATLAB Draw a polygon from line equation

    I'm trying to draw a quadrilateral in matlab. I has 4 line equations. Can i draw that polygon, using line equations ? I knew MATLAB can draw polygon from coordinates, but i don't want to use it.
  13. twoski

    I Getting Points Inside Of A Polygon

    Let's say we have a somewhat complex polygon shape. We know the list of 2d vectors that make up its border loop. Now let's say i want to create a list of points within this polygon. Currently i am using ray traces to create a grid of points that fall within the polygon. I'm certain there is...
  14. 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?
  15. YoungPhysicist

    I Area divided by linear function

    The original problem for anyone that can read Chinese: https://zerojudge.tw/ShowProblem?problemid=b221 The problem defines a convex polygon with multiple points located in the first quadrant and the required task is to find a linear function y = ax that can spilt the polygon into two parts each...
  16. H

    MHB Law of Sines in Polygon Construction - Goals & Benefits

    In the link: https://www.maa.org/external_archive/joma/Volume7/Aktumen/Polygon.html Why the law of sines is permitted here? What are the goals of the author to use it?
  17. H

    MHB Understanding the Relationship between Angles & Diagonals in a Polygon

    There a relationships between angles to diagonals in a polygon?
  18. CDL

    Point Charges on a Polygon with another Charge in the Middle

    Homework Statement Suppose we have a regular n-gon with identical charges at each vertex. What force would a charge ##Q## at the centre feel? What would the force on the charge ##Q## be if one of the charges at the vertices were removed? [/B]Homework Equations Principle of Superposition, the...
  19. Vital

    I Reading the vertical data in a frequency histogram polygon

    Hello. Please, take a look at the screenshot from the textbook. They say in the textbook that there are in total 48 data observations, 20 of which lie in the interval 0 - 2, and 6 lie in the interval 2 - 4. Yes, both 20 and 6 are more or less clear on the graph, but how did they come up with 48...
  20. Z

    Interior and Exterior Angle of a Polygon: Formula not working

    Homework Statement How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle? Homework Equations Interior angle of a polygon: ((n-2) * 180)/n Exterior angle of a polygon: (360)/n The Attempt at a Solution ((n-2) * 180)/n = 8 *...
  21. J

    I Choosing points from a set that produce the largest polygon

    I am trying to find the most efficient way to select points on a 2d plane from a set that maximizes the area of the of the shape they define when joined together. The points are all paired (sharing the same A->B vector), with these pairs also appearing mirrored about the origin. Here is an...
  22. A

    Book Question -- velocity polygon and corelis acceleration

    Can someone please tell me what book is about velocity polygon and corelis acceleration? Thank you
  23. 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...
  24. M

    Proving length of Polygon = length of smooth curve

    Homework Statement The problem statement is in the attached picture file and this thread will focus on question 7 Homework Equations The length of a curve formula given in the problem statement Take a polygon in R^n as an n-tuple of vectors (a0,...,ak) where we imagine the vectors, ai, as the...
  25. A

    MHB Algebra help - a race around a regular polygon

    Bert and Ernie are running around a regular polygon with x sides, all of length 12m. They start from the same point and run in opposite directions. If Bert is twice as fast as Ernie, how far will Ernie have traveled when they meet?
  26. F

    Simplifying Exterior Angles in a Polygon

    Homework Statement Homework Equations sum interior angles (n-2)*180 angles of a quadrilateral: a+b+c=d = 360 [/B] The Attempt at a Solution What do you do with the exterior angle? 80+130+a+x=360
  27. J

    Moment of Inertia for Simple Polygon

    I am trying to determine if a function I have is working correctly. The function computes the moment of inertia for a polygon. To test the function I created a polygon that has 4 points and represents a square of 4 units by 4 units. The answer I am getting is 32. I believe the answer should be...
  28. 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...
  29. Estanho

    CCW or CW ordering of the points of a generic polygon

    Hi, Suppose I have a data structure that models a polygon, by storing all the nodes of the given polygon, and their connections (i.e. edges). I understand that sorting a set of nodes that may form a concave hull is ill-defined, as there could be many polygons that can be formed with those. But...
  30. N

    Calculating Diagonals of a Polygon: What are the Values of p and q?

    Homework Statement A polygon with n sides has a total of 1/p . n . (n-q) diagonals, where p and q are integers. (i) Find the values of p and q. Homework EquationsThe Attempt at a Solution Can someone just help me to start on this? I know that q = 3 and p = 2 but how?
  31. S

    Circle or polygon for charged particle in magnetic field....

    ...perpendicular to its path? OK; let's say you have any charged particle moving perpendicular to a magnetic field; does it describe a gigantic polygon or a perfect circle? I think it's a polygon; the particle absorbs a "quantum of force" from the magnetic field, so to speak, and changes...
  32. M

    Polygon sine functions? what is this?

    Hi! I was wondering how I could find the equations for the bottom two functions. I understand that the amplitude is not constant like that in the circular sine function--could someone please help me out? Thanks!
  33. T

    How can I use matrix algebra to rotate a polygon and calculate new coordinates?

    Let's say I have a 5 vertex polygon drawn on a x/y graph. Each of those vertices have an x/y coordinate. Now let's say I want to rotate the polygon 25 degrees clockwise and calculate the new coordinates of each vertex. How can I do that so that by knowing the original coordinate I can do a...
  34. V

    Determining the Velocities of a Polygon in a Curve

    Hello, everyone! My question is really simple, in fact I even feel a bit ambarrassed of asking it. :x Imagine that a car is making a constant radius turn, of a given radius R. For the purposes of this question is enough to say that the car may be thought as an isosceles trapezium, or even as a...
  35. trytodoit

    How to derive the formula for moment of inertia of polygon?

    Sorry to bring this question up again. @aridno provides a nice formula of the moment of inertia I about the centroid in https://www.physicsforums.com/threads/calculating-polygon-inertia.25293/ as: $$...
  36. K

    Conformal mapping from polygon with circle segments

    I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a circle segment. The closest I can find is the Schwarz-Christoffel mapping. Anyone has any tips?
  37. T

    Define boundary conditions of a polygon in a unit square cell

    Hi, I am wondering as to how to define the boundary condition for a shape in a unit cell. Just imagine that the shape is the hole for the unit cell. Hence, for a constant thickness on the untextured boundary, thickness is, let's say C, then for the circle, it's C+depth. For example for...
  38. A

    Area of a polygon- using numerical integration

    Hi, I need to calculate area of an irregular polygon which can be of any complex shape numerically i.e. using numerical integration techniques. Please can anyone suggest any reference material / best way of going about this efficiently? Akash
  39. S

    Circles vs an infinitely n-sided polygon.

    Suppose you have a square, and you simply start increasing the number of vertices and edges proportionally, all the way to infinity. What, exactly, distinguishes this infinitely sided polygon from a circle? Logically, an infinitesimal edge would be like a point on a circle, although I...
  40. I_am_learning

    Area of a general n-sided polygon

    Finding the area of an irregular polygon with n side is quite easy when we are given the length of all of the n sides and the length of (n-3) specific diagonals. This way, we get (n-2) triangles whose areas can be calculated using Heron's formula and then added up. But what if the length of...
  41. J

    MHB Number of quadrilatera in polygon

    Number of Quadrilateral that can be made using the vertex of a polygon of 10 sides as there vertices and having Exactly $2$ sides common with the polygon My solution: first we will take $2$ sides common is $=10$ ways now if we take two sides as $A_{1}A_{2}$ and $A_{1}A_{3}$ then we will...
  42. D

    Finding the moment of inertia of a 2D polygon.

    Hi everyone, Is there a general method for finding out the moment of inertia of an irregular convex 2D polygon given the coordinates of its vertices? I have thought of one possible method: Split the polygon into multiple triangles and find the moment of inertia of each triangle around the...
  43. A

    Polygon Collision calculate velocity and angular velocity

    Hello, I am trying to simulate collisions between polygons with opengl. The weight, velocity (as a 2d vector), the angular velocity and the collisionpoint of the colliding polygons are provided. How can I calculate how the angular velocity and the velocity are changed? I do know how i...
  44. M

    Charges in a polygon distribution

    Homework Statement 13 equal charges q are placed at the corners of a polygon and as on a clockface, what is the net force on a test charge at the center Q? I assume the charge at the center is positive and the charge on the polygon so they repel Homework Equations (1/4piEo)(qQ/r^2) The...
  45. D

    Fundamental polygon of a Mobius strip

    Hey I am having a little bit of difficulty. The classification theorem for 2 - manifolds tells me that every 2 -manifold has the following representation: 1) connect sum of n-tori 2) connect sum of n-projective planes 3) a sphere Now, using Massey's book there is a very algorithmic...
  46. Sudharaka

    MHB Edgar's Question from Facebook: Convex Polygon

    Edgar from Facebook writes: The sum of the measures of the interior angles of a convex polygon is ten times the sum of the measures of its exterior angles. Find the number of sides of a polygon. Hello could you please help me to solve this problem?
  47. R

    Summing the Mountains and Valleys of a Regular Polygon

    In every top of a regular polygon with 2n tops there is written an integer number so the numbers written in two neighboring tops always differ by 1 ( the numbers are consecutive ) The numbers which are bigger than both of their neighbors are called ”mountains” and those which are smaller than...
  48. P

    Algorithm for testing intersection of point and compound polygon

    I'm trying to find a reasonably fast method for testing whether or not a point (x,y euclidean coordinate system) lies inside a (preferably convex, concave or complex - though different methods for each would be OK) compound polygon with edges consisting of line segments, arcs and/or elliptical...
  49. E

    Parametrization of a regular planar polygon with an arbitrary number of sides

    I was wondering if anyone knew of a common technique for parametrizing a regular polygon with an arbitrary number of sides. I figured such a problem would be easy or at least be well documented online, but that doesn't seem to be the case. I started by assuming that the polygon was centered...
  50. Saitama

    Probability, Diagonals of a convex polygon

    Homework Statement In a convex polygon of 6 sides, 2 diagonals are selected at random. The probability that they intersect in the interior of the polygon isHomework Equations The Attempt at a Solution There are 9 diagonals in a polygon of 6 sides. Therefore the total cases are 9C2. But how...