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

learn more… | top users | synonyms

1
vote
0answers
14 views

Does it make sense to compare sets (polygons) with different dimensions?

In the context of integer programming, I am considering 3 different linear models for a given problem. The goal is to determine which formulation is the tightest, that is, the one that gives the least ...
-1
votes
2answers
38 views

How would the volume of a frustum with irregular polygon area be calculated?

I want to calculate the volume of this shape, it's basically a frustum with an irregular polygon base. The bottom area $A_1$, the height of the frustum shape $h$,the sideways distance between $A_1$ ...
2
votes
1answer
484 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
10
votes
3answers
107 views

Closed form for the area of a convex cyclic n-gon, given the set of edge lengths

Let's say we are given a set of positive reals, and we're told that these are the edges of a convex cyclic $n$-gon, and we must compute it's area. For $n = 3$ there is the famous Heron's formula: ...
1
vote
1answer
18 views

Determine whether a polygon is convex based on its vertices.

We have a polygon $A_1A_2\ldots A_k \subset \Bbb{R^2}$ with the coordinates: $$A_1 = (x_1, y_1)$$ $$A_2 = (x_2, y_2)$$ $$\vdots$$ $$A_k = (x_k, y_k)$$ Is there any way to determine whether or not ...
0
votes
2answers
74 views

Prove that $\frac{ a^2 +b^2 +c^2}{d^2}$ is always greater than $\frac{1}{3}$

Prove that if $a,b,c$ and $d$ are the sides of a quadrilateral ,then the value of $\frac{ a^2 +b^2 +c^2}{d^2}$ is always greater than $\frac{1}{3}$ Could someone please give me hint to solve this ...
0
votes
1answer
36 views

Center of gravity of right angle trapezoid vs rectangle

Could you please help me to find the center of gravity of a trapezoid and a rectangle with the following measures? Where is the center of gravity in these two shapes? Trapezoid: a: $20$cm b: $17$ ...
1
vote
0answers
20 views

Maximum number of regular polynom projections of a polyhedron

It is trivial that a cube has both a square and a regular hexagonal projection. We can also easily construct a polyhedron with three perpendicular projection, which are different regular polygons. ...
1
vote
0answers
89 views

Progressive packings in a convex shape

Take a shape, and scale it by 1 to $n$. For a tiny set of tightly related shapes, such as isosceles right triangles with shortest sides 1 and sqrt(2), scale the set of shapes by 1 to $n$. What is ...
0
votes
1answer
71 views

Regular polygon made from the midpoints of the sides of a polygon

For what natural numbers n is it true that whenever the midpoints of the sides of a convex n-sided polygon $K_n$ form a regular n-gon, then $K_n$ itself is also a regular n-gon? I know it is true for ...
2
votes
2answers
39 views

Polygon with curved sides, and higher-dimensional generalizations

I am trying to find references about generalizations of polygons with non-straight sides. I am interested in both the convex and non-convex cases, and particularly in polynomial boundaries, and ...
0
votes
2answers
22 views

Are there alternatives to polygons in mathematical (computational) modelling?

So polygons are pretty standard in computer graphics, but from a mathematical perspective, one'd expect something more refined and sophisticated to be possible right? Polygons are not very ...
1
vote
0answers
18 views

create polygon section with equal sides

I have to create essentially these sections of a polygon. I have width(W) and height (H), and number of sides (3 on left abc and 4 on right image ABCD) I need each side to be equal. How can I ...
0
votes
1answer
365 views

Number of sides a regular polygon has.

The question is "Both tile A and B are regular polygons. Work out the number of sides A has." For this I put B is equilateral ∴ all angles are 60. However, I have no idea where to go from ...
1
vote
3answers
8k views

How do you solve the area of a trapezoid using diagonals

The height of a trapezoid is $10$ cm. The lengths of the two diagonals of the trapezoid are $30$ cm and $50$ cm. Calculate the area of the trapezoid. On the homework I solved this using ...
4
votes
2answers
45 views

Average Perimeter With n Points on the Unit Circle

A couple days ago, a friend challenged me to solve a problem: You have N vertices, each randomly placed on the edge of a unit circle. What is the formula (given N) that yields the average perimeter ...
0
votes
3answers
62 views

Is there a regular hexagon with integral corners?

I'm looking for a regular hexagon in $\mathbb{R}^2$, whose corners are integral, i.e. the coordinates are integers. The hexagon cannot lie "flat" (with upper and lower line segments horizontal), ...
15
votes
2answers
409 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
1
vote
1answer
65 views

Find if 4 lines form a Quadrilateral in 2D space

How can I know if 4 lines form a Quadrilateral in 2D space? And how would I obtain the corners? (in clockwise order starting with the top left corner) Note that lines are formed by 2 points in my ...
3
votes
2answers
51 views

Largest four line segments of polygon

I have some polygon (see darkblue contour): It consists of very small segments, pixel by pixel, so angles differ although they seem to be the same. Visually we see 4 large line segments. How can I ...
0
votes
1answer
23 views

Formula for vertices of a Polygon with only 1 vertex at the top and y-axis symmetric

I'm trying to find the formula for the vertices of a polygon with n-sides such that there is always only 1 vertex at the top and the polygon is symmetric with respect to the y-axis... so generally ...
3
votes
2answers
60 views

Prove that there does not exist a $n$-regular polygon $(n\ge 4)$, such that its sides and diagonals are all integers.

Prove that there does not exist a $n$-regular polygon $(n≥4)$, such that its sides and diagonals are all integers. Maybe a famous problem, but I don't know how to solve that.
2
votes
4answers
2k views

Prove using integration that $polygon → circle\space \text{as}\space number\space of\space sides → infinity$ [closed]

Say we have a regular polygon $s$, with number of sides $n$: Is there a way to prove that as $n → ∞,\space $then $s → circle$ using integration?
0
votes
0answers
13 views

What is the boundary component of the polygon?

A polygon has the edge equation, R(f)P(b^-1)Q(d)S(e^-1)R=1 where R, P, Q and S are the vertices and f, b, d and e are the edges. What would the boundary component be. I know the boundary edge is f, b, ...
0
votes
1answer
23 views

Minkowski sum and Polygons

The problem:.. Given two convex polygons $A$ and $B$, we can define Minkowski sum as A + B = {a + b: a $\in$ A, b $\in$ B}, where $a + b$ vector sum. Prove that: every vertex $p \in A + B$ is a ...
19
votes
7answers
806 views

Number of ways to connect sets of $k$ dots in a perfect $n$-gon

Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices in a given perfect $n$-gon such that no two lines intersect at the interior of the $n$-gon and no vertex remains ...
1
vote
1answer
54 views

Minkowski sum and vectors

Problem: Given two convex polygons A, B, we can define Minkowski sum, as A + B = {a + b: a $\in$ A, b $\in$ B}, where a + b vector sum. Prove that: for every external perpendicular u to an edge of ...
0
votes
1answer
11 views

Degree measure of multiple polygons

I made this design on the Desmos calculator, and I was wondering what the quickest way was to find the degree measure of each individual angle. What I know so far: The measures of each of the ...
2
votes
1answer
41 views

Proof for Centroid Formula for a Polygon

I was reading a paper and I found this formula for the centroid of a polygon in terms of its coordinates but no proof was given. $C_x =\frac{1}{6A} ...
6
votes
0answers
81 views

Counting Regular polygons in Complete Graphs

The figure shows the correct $24-$gon, which held all the diagonals. a) Find out how we got right triangles and squares (question for arbitrary $n$)? b) How this problem can be generalized (if it is ...
2
votes
1answer
46 views

Pentagonal tiling

I am currently working on a research project in my last year of high school. For this paper we are discussing Eschers tesselations, both in the euclidian and the non-euclidian plane. At the moment I ...
0
votes
2answers
28 views

Area enclosed by a polygon

I did some work in the area of mensuration and came across an interesting concept/formula. The formula states that For a polygon having vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3),\dots(x_n,y_n)$, the ...
1
vote
1answer
38 views

Regular polygon inscribed in a unit circle

Given a point $P$ on the circumference of a unit circle and the vertices ${A_1},{A_2}, \ldots ,{A_n}$ of an inscribed regular polygon of $n$ side. Prove that $P{A_1}^4 + P{A_2}^4 + \cdots + ...
0
votes
1answer
73 views

Value Of $\pi$ obtained using limits!

What i thought was simple, a circle can be formed by increasing the number of sides of regular polygon( like pentagons, hexagons, etc ) up to infinity by keeping the distance between the center and ...
3
votes
1answer
63 views

Predicting Spirals

I am currently in the process of analyzing a polyspiral, a spiral where each successive length drawn is increased at specified increment at the same angle. *Please note the angles selected are the ...
0
votes
1answer
33 views

There exist three consecutive vertices A, B, C in every convex n-gon with n≥3, such that the circumcircle of triangle ABC covers the whole n-gon

From Problem Solving Strategies by Arthur Engel: Problem to prove: There exist three consecutive vertices $A$, $B$, $C$ in every convex $n$-gon with $n \ge 3$, such that the circumcircle of triangle ...
2
votes
2answers
75 views

Placing circles inside of a regular polygon.

Alice and Bob play the following game: on a table there is a regular $n$-gon. On each person's turn, they are required to place a circle of radius $r$ fully in the interior of the $n$-gon such that it ...
1
vote
1answer
21 views

Calculating radius of circles which are a product of Tangent Intersections using a Regular Polygon

Introduction Lets have a regular polygon of $n$ sides inscribed in a circle of radius $H$, then construct tangents between the circle and each point of the polygon and draw new circle(s) trough the ...
0
votes
2answers
21 views

For the parallelogram, Prove $XY=CD$

$ABCD$ is a parallelogram. The bisectors of $\angle A$ and $\angle B$ meet BC and AD at X and Y respectively. Prove that $XY=CD$? Please give me some hint to prove it. I can't initiate the problem so ...
1
vote
1answer
20 views

Why reflection and rotation are sufficient operations in dihedral group?

I know a bit of elementary group theory but please ignore dihedral group in the title and let's make it simple enough so a high student can read the question and the answer(s)... Suppose we have a ...
1
vote
1answer
30 views

Why the number of symmetry lines is equal to the number of sides/vertices of a regular polygon?

Considering the/a definition of a regular polygon from Wiki : In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all ...
2
votes
0answers
25 views

Area covered by fixed perimeter around polygon.

Suppose I have a polygonal field with a post at each vertex and a non-extensible rope threaded through each post around the perimeter but with some slack. How can I determine the perimeter of the area ...
3
votes
0answers
47 views

Centroid and circumcenter — how close?

Suppose $R$ is some planar region, bounded by a curve. Let $C_1$ be the centroid of $R$, and let $C_2$ be the center of the "circumcircle" (the smallest circle enclosing $R$). Intuitively, it seems ...
6
votes
1answer
63 views

Calculating radius of circles which are a product of Circle Intersections using Polygons

Lets say you imagine a circle with the radius $R$ and you inscribe a regular polygon with $n$ sides in it, whose side we know will then be: $$a=2R*sin(\frac{180}{n})$$ Then you draw a set of circles ...
1
vote
1answer
49 views

Interior angles of irregular quadrilateral with 1 known angle

I have the measurements of the four sides of an irregular polygon and I need to find out the size of each interior angle. I know the sum of the angles is 360 degrees but because it's not a regular ...
0
votes
1answer
12 views

consider the following statements regarding the smallest interior angle of a n sided polygon with perimeter n units and with maximum area?

let(f) be the relation defined by f(n) = The smallest interior angle value of the n sided polygon with perimeter n units with maximum area, for each positive integer n(>2).which of the following are ...
2
votes
2answers
38 views

Why am I getting the wrong formula for the area of a dodecagon?

More likely than not, I'm just making a simple algebraic mistake, but I can't seem to find it and so I would like some help. Divide a (regular) dodecagon into $12$ congruent isosceles triangles with ...
2
votes
3answers
35 views

Triangles with no common side in a polygon

There are n sides of a polygon(where $n>5$). Triangles are formed by joining the vertices of the polygon. How many triangles can be constructed with no side common to the polygon? My try: Total ...
5
votes
3answers
112 views

What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational ...
1
vote
3answers
80 views

Is there a theoretical (or practical) definition of $n$-gon, for $n < 0$?

Background This is purely a "sate my curiosity" type question. I was thinking of building a piece of software for calculating missing properties of 2D geometric shapes given certain other ...