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. J

    Problem expanding algebraic function via Newton polygon

    Hi, I've run into a problem with expanding algebraic functions via Newton polygons. Consider the function: f(z,w)=a_0(z)+a_1(z)w+a_2(z^2)w^2+\cdots+a_{10}(z)w^{10}=0 and say the degree of each a_i(z) is ten. Now suppose I wish to expand the function around some ramification point of the...
  2. O

    Finding E on the Z Axis: A Differential Approach

    Imagine you have a regular 13 sided polygon with charges distributed on every corner of the polygon. What would a test charge experience in the center? The answer to that was a 0 net force (which makes some intuitive sense to me due to the symmetry of the polygon). I understand that if we...
  3. Blandongstein

    MHB Find Area of Polygon: Simpler Method?

    Find the area of the polygon formed by the points (3,5), (5,11), (14,7), (8,3), and (6,6). I can find the area of the polygon by dividing it into 3 triangles and then finding area of each triangle separately. I want to know if there is any simpler way of doing this.
  4. C

    MHB Minimum area of an odd number-sided, equilateral polygon with side lengths of 1 unit.

    Suppose you look at all of the equilateral (non-self-intersecting) polygons** with an odd number of sides, and each side length is equal to 1 unit. For examples, the polygon with the fewest number of sides in this group is the equilateral triangle, and then the next one is an equilateral...
  5. F

    Relationship between funicular polygon and bending moment diagram

    Hello, I'm a civil engineering student. I learned how to contruct a funicular polygon and bending moment diagrams in two different subjects and I realized how close looking the two are, so I wondered if there was a good explanation to relate the two. I can feel it's kind of the same as the...
  6. N

    Product of Diagonals of Regular Polygon?

    So any help would be really appreciated! I really have no idea where to start, and I can use any help. So essentially the problem is we have a regular polygon P inscribed in a unit circle. This regular polygon has n vertices. Fix one vertex and take the product of the lengths of diagonals...
  7. A

    How to get the new Coordinate of an Polygon at angle X

    Let say I have a triangle(polygon). I know all the co-ordinates of all points(x1,x2,x3,y1,y2,y3). Let say the polygon is inclined(at x3,y3) and it's angle is 30 degree. How to get the point x4, y4?
  8. N

    Interior angles of polygon on a sphere

    Hi can anyone help me out with finding the interior angles of a pentagon on a sphere. I know two of the interior angles already and I know all the angles that correspond with the arc lengths of the sides of the pentagon. How do I find the other three interior angles? Thanks
  9. A

    Divide convex polygon into 4 equal areas

    Homework Statement Show that it is possible to cut any convex polygon into 4 pieces of equal areas by using two cuts perpendicular to each other. Homework Equations None, it's just a proof I found on the back of my book. The relevant chapter is Continuity, the maximum principle, and...
  10. Z

    Algorithm for cutting a polygon into specific shapes

    Hello everybody, I have a shape (polygon) made up from pixels (squares) similar to the image below. I need an algorithm to cut it into "lines" i.e. shapes of 1x[1..20] pixels. The lines should not be necessarily straight, but they should fill the entire area. Any ideas on where to start...
  11. T

    Summing n-number of Terms to Find the Area of a Polygon

    Homework Statement Let C be the line segment connecting the points (x1,y1) and (x2,y2). More over let the line integral over C of (x dy - y dx) = x1y2 - x2y1. Suppose the vertices of a polygon, listed in counter-clockwise order, are (x1y1), (x2y2), ... , (xnyn). Show that the area of the...
  12. N

    Ordering Line Segments to form a 2D Polygon after slicing a 3D Tri Map

    I have a 3D shape described by a triangulation map i.e. a map between the vertices to the faces of the shape which are all triangles. I then sliced the shape by a plane and computed the intersections of the plane and the triangle faces. Each triangle face that intersects the plane, will have...
  13. B

    Infinite summations: area of polygon

    Prove that the area of the polygon with vertices at (-1,0), (-1+2^(-n), 1-(-1+2^(-n))^2), (-1+2(2)^(-n), 1-(-1+2(2)^(-n))^2),..., (1,0) is 1 + 4^(-1) + 4^(-2) + ... + 4^(-n). I tried using the formula for the area of a polygon but could not get the answer. Now sure how to prove this.
  14. S

    Exploring the Relationship Between Ratios of Sides and Areas in Regular Polygons

    Homework Statement 1. In an equilateral triangle ABC, a line segment is drawn from each vertex to a point of the opposite side so that the segment divides the side in the ratio 1:2, creating another triangle DEF. a. What is the ratio of the area of the two equilateral triangles? b. Check the...
  15. S

    Polygon Of Forces To Demonstrate Equilibrium

    Homework Statement Use the polygon of forces to demonstrate that a 200 kg crate on a 25o incline is in equilibrium on the slope. Homework Equations See attached The Attempt at a Solution Please see attached for my attempt, I think I am in the right ball park, just need to know what...
  16. L

    Biot Savart polygon with n edges

    Homework Statement I want to calculate the magnetic field with Biot Savart in the given drawing in the point P=(0,0,0) Homework Equations Biot Savart The Attempt at a Solution I have already problems in parametrizising the conducter loop. Can anyone give me some hints on...
  17. A

    Opposite of Polygon: Open Shapes & Names

    Sorry if this seems elementary but what shapes are the opposite of polygons (closed by line segments) and what do you call them? Sorry if my question is phrased weird.
  18. G

    Normal polygon area without trig functions

    Here's an interesting problem: How can you find the area of any normal polygon with x sides (or corners) that is inscribed in a circle of radius 1? No trig functions, or things like e or π (Pi), or infinite series, are allowed. If possible, try to avoid summation notation as well, but that might...
  19. B

    Prove Polygon of Forces Not Necessary for Equilibrium

    If any number of forces acting on a point is in equilibrium then the forces does not necessarily form a polygon of forces? How can we prove it?
  20. J

    Line intergal of a polygon

    Homework Statement Evaluate Integral y^2 dx + (xy - x^2) dy over the given path C (0,0) to (2,4) the polygonal path (0,0), (2,0), (2,4) (All one question) Homework Equations integral of h (dot product) dr over C The Attempt at a Solution I realize I have to parametrize the...
  21. Z

    No. of quadrilaterals in a polygon

    Homework Statement Quadrilaterals are formed by using the vertices of a convex polygon of 24 sides. The number of quadrilaterals having atleast one side of the quadrilateral in common with the side of the polygon is ? The Attempt at a Solution We can find out the total no. of...
  22. S

    Resultant of thirty vectors of a polygon

    equation, square of resultant, R2 = Rx2 + Ry2 [b] interior angle of a polygon = (n-2)* 180°/n = (30 - 2)* 180°/30 = 168° components on X axis: Ax1 = 20* cos 168° Ax2 = 20* cos (168°+168°) . . . . Ax30 = 20* cos...
  23. S

    Polygon Detection for Bay Selection and Vertical/Horizontal Line Drawing

    1. I have a problem like this: I. As you see (in the attachment) each of the closed polygons (as in the attachment) is called a BAY II. The logic I need is: The user of my program will select a bay (with a mouse). And after selection he would want the program to draw some vertical (or...
  24. M

    What is the relationship between similar regular polygons?

    Homework Statement I am reading this trig book and it is saying that if both are reg polygons ( I am assuming they would have to have the same sides) that they are ratios of one another... I would like to read more on this so I understand it better... is there a link anywhere that someone can...
  25. R

    Polygon of force diagram inclined plane

    Homework Statement Use the polygon of forces to demonstrate that the crate is in equilibrium on the slope Homework Equations No friction 1962 N crate sitting on a inclined plane 25 degree angle. The Attempt at a Solution I have drawn four forces mg = 1962 N ( 200 kg * 9.81)...
  26. J

    Interior angles of a regular polygon

    The number of sides of two regular polygons are in the ratio 5:4 and the difference between their interior angles is 6 degrees.Find the number of sides of the two polygons. I forgot the relation between interior angles and the number of sides of a regular polygon.Can anyone help me to figure...
  27. Z

    Figuring Out Clockwise Polygon Fill in a Program

    I don't know exactly how to explain this, but I'll try my best: Let's say I have a set of points (P1, P2, P3...Pn), that are the vertexes of a n-sided polygon. As would be expected, the polygon is drawn simply by connecting the points in order (P1 to P2, P2 to P3, Pn to P1). This is the...
  28. S

    Efficient Computation of Complex Polygon Areas?

    Hello. Nice to be here. If I may, I would like to inquire about the enclosed area of complex polygons. Is there a general formula that will work for these and reduce/cancel out partly for simple/non self-intersecting polygons for a correct enclosed area of theirs as well? I need to compute...
  29. 1

    Solve Polygon Problem: 1:2 Ratio & 3:4 Sum

    polygon problem (?!) Hi there!:smile: I have got an problem here.It's a simple one.Not much to think about "The ratio between the numbers of sides of two regular polygons is 1:2 and the ratio between the sum of their 3:4.Find the number of sides in each polygon" It appears easy and is too...
  30. W

    Prove Symmetry Group of Regular Polygon Has 1 & 2 Dim Irreducible Reps

    how to prove that the symmetry group of a regular polygon has only 1 and 2 dim irreducible representations?
  31. qspeechc

    What shape has the largest area for a fixed perimeter?

    Hello everyone. While I was waiting for my computer program to run, I occupied myself with this little problem. For a fixed perimeter, which regular polygon (or any closed shape in the plane) has the largest area? The answer is the circle (I guess), if we regard it as an infinite-sided...
  32. M

    Formula/Algorithm to apply force to an arbitrary point on a polygon

    I am trying to code a 2D rigid body physics engine in Java, and I am having some trouble figuring out how an object will move and rotate if the force is off-center. Basically, what I can't seem to figure out is a way to find out the x, y, and rotational acceleration when a force is applied at...
  33. C

    Split simple polygon into Monotone pieces

    Hi all! I develop an application in computer science that shows the execution of an Algorithm that triangulates a simple polygon. The first thing I have to do is to transform the given Polygon into monotone pieces, in order to do that I have to figure out the interior angles of the simple...
  34. C

    Show set is a polygon connected set?

    Homework Statement Show that the complement of the set S = {all (x, y) with x and y rational numbers} is a polygon connected set. Is it an open set? Homework Equations The Attempt at a Solution The complement of S = {all (x, y) with x not rational or y not rational} Let the...
  35. icystrike

    Finding the Minimum Number of Sides for a Rotating Regular Polygon

    Homework Statement A floor tile has the shape of a regular polygon. If the tile is removed from the floor and rotated through 50◦ it will fit back exactly into its original place in the floor. The least number of sides that the polygon can have is? I don't know what are the theories that i...
  36. Q

    How to Plot a Polygon on X-Y Graph Knowing the Radius

    Homework Statement I want to plot a Regular Polygon of many sides on a X-Y graph where I know the number of sides and the radius. I would like a method to calculate the position of the corners of this shape without using compass/ruler. If there was an algorithm that goes all the way around...
  37. K

    Complex- area enclosed by a polygon

    Homework Statement Recall that the area enclosed by the polygon with vertices z1,z2,z3,...,zn is 1/2I(z1conguatez2+z2congugatez3+...+zncongugatez1) Show that the area enclosed =1/2I\Sigmazkcongugate(zsub(k+1)-zk). Interpret this sum as part of the approximating sum in the definition of...
  38. T

    Circle and Polygon Properties

    Homework Statement http://i41.tinypic.com/35mno6v.jpg 2. The attempt at a solution So far, I found out that angle 4 and 5 is 55, because angle D is 110.. but I don't know if that's right. Please help!
  39. E

    How do you find the moment of inertia of a polygon?

    I'm working on an engine right now, and I'm having trouble calculating the moment of inertia for a polygon. Is there any way to easily do this without decomposing the polygon into triangles? edit: I've looked at the wikipedia page with examples on the subject...
  40. J

    Can a contour integral be used to detect complex polygon deformations?

    Hi, I am working on a simulation code that simulates the deformation of sand grains in 2D. The sand grains are modeled as simple polygons. However, during the simulation the grains can deform to create non-convex vertices. Further more, when deformation becomes extreme, there is a possibility...
  41. N

    Inner-radius of a (convex) polygon

    Hi all, I want to know whether it is correct that every convex polygon has an inner-circle (& hence an inner-radius). I think it is only possible for triangle and for regular polygon. Am I right? If there is any convex N-gon having sides a_1,a_2,...,a_N which has an incircle, then...
  42. W

    Rectangular Resolution and Polygon Theorem

    1. Find the resultant of two forces of 40 lbs. and 50 lbs. acting at an angle of 60○ between them. 2. Three forces of 30 gms, 50 gms, and 60 gms respectively act at an angle of 120○ from each other. Find the resultant by rectangular-resolution (a) by making the 30-gm force lie on the x-axis...
  43. Mentallic

    How can I improve my proofs in geometry?

    Geometry is arguably my weakest link in mathematics. The answers just don't "hit me" in geometry like some other sections of math do. When trying to prove something in a polygon, such as congruence of triangles made by segments etc. I find it difficult since the equal sides/angles aren't...
  44. T

    Get Polygon Side Length From Radius

    I'm doing some programing and I am wondering how I can find the length of a Polygon's side (variable number of sides) from the radius (distance between center and one corner). Thank you _Zachariah
  45. I

    Segments of a polygon

    Hi all, I have this math problem where an equilateral triangle has each of its sides divided into ratio 2:1, and a smaller equilateral drawn within it from the intersecting lines. There is also a similar problem, but relating to a square. My objective is to figure out a relationship between...
  46. S

    Moment of inertia for Concave Polygon

    [SOLVED] Moment of inertia for Concave Polygon While working on a simulation I ran into this problem. I'm trying to calculate the moment of inertia for a concave polygon. The polygon is made of N vertices (Also the edges are straight lines). I've done a bit of researching however I've only...
  47. S

    Group of symmetries on a regular polygon

    So i began reading up on some group theory and I came across an interesting question, what is the order of the group of symmetries on of a n-sided regular polygon? with a square it's 8, triangle it's 4. I feel like I'm missing something with the pentagon because I'm only finding these: the 5...
  48. E

    Some statics confusion - Vectors & Polygon of forces

    Hi all, I am a little confused with a supposedly simple statics concept. The topic was on vector forces and the polygon of forces with respect to a static mechanics problem. The text i was reading was a little confusing: The text also shows a diagram both spatial and a 2D polygon of...
  49. C

    Discover the Mysterious Behavior of Fluid in Polygon Shaped Vortices | Phys.org

    http://www.physorg.com/news66924222.html%22 Why does the fluid act like this? I'm truly stumped... I would read the published results paper, but I can't pay for it.
  50. P

    How was the angle sum of polygons derived?

    We all know that for the angle sum of the external angles of a non-concaved polygons is 360. How is/was this derived...
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