Skip to main content

Questions tagged [polygons]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
25 views

Rational quantities associated with a bicentric heptagon

For odd reasons of my own, I would like to create a math puzzle involving a bicentric heptagon (i.e., a 7-sided polygon which has both an inscribed and a circumscribed circle). The puzzle is to be ...
tuna's user avatar
  • 547
-4 votes
1 answer
97 views

Why is the intersection point of the diagonals of a regular polygon always the center?

Drawing the long diagonals of an even sided regular polygon, why is the intersection point always equidistant from the vertices? This is an assumption in many problems, but I have not seen it ...
Problem_Solving's user avatar
0 votes
0 answers
26 views

Polygon Boundary in 3D Space

A simple polygon (interior included) is a manifold with boundary being homeomorphic to a closed disk. Its (manifold) boundary is a non-intersecting closed polygonal chain. Viewed as a subset of the ...
Wipetywipe's user avatar
-3 votes
0 answers
56 views

How do you prove that a specific shape in a regular octagon is a rectangle? [closed]

In regular octagon ABCDEFGH, how do you prove that ADEH is a rectangle? This is assumed in many problems that find the area of an octagon. Since AD and EH are symmetrical diagonals, they are equal, ...
Anonymous's user avatar
-3 votes
1 answer
35 views

Find the value of the interior angles of a polygon [closed]

I have coordinates of points $(x, y)$. By connecting these points we get a polygon in which I have to get values of its internal angles. For example points = $[3,1], [3,3], [1,3], [3,5], [7,5], [7,1]$ ...
Paul's user avatar
  • 97
1 vote
0 answers
20 views

Circular random walk to generate a Polygon

I am trying to generate a set of points distributed in such a way as to give a "rough circle" sort of shape. The points should not deviate too far from neighboring points, with larger jumps ...
Anthony Khodanian's user avatar
2 votes
0 answers
33 views

EdExcel iGCSE Higher Maths PAPER 2, Q24 - possible mistake?

I believe there is a mistake in the following question which was asked at the end of this year's iGCSE maths paper 2 in the UK. Please read through my solution or attempt it yourself and see if you ...
Penguinking14's user avatar
6 votes
2 answers
551 views

Mismatching Euler characteristic of the Torus

Why is it that when I try to compute the Euler characteristic for the Torus using a drawing like the following , then the number that I get is not the number that the Torus should have? Which is $0$? ...
Tutusaus's user avatar
  • 657
0 votes
1 answer
38 views

Create a regular n-gon from sides and bounds width [closed]

I am trying to create a regular polygon with an arbitrarily number of sides with the starting parameters: number of sides, width of bounding box. And the polygon should have an edge at the base. For ...
Barreto's user avatar
  • 131
0 votes
0 answers
23 views

Prove that the Hausdorff distance and Area metric are not equivalent on the set of all bounded plane polygons.

Prove that the area metric, $d_{\Delta}$, is not equivalent to the Hausdorff distance between two sets. The book and definitions are here [1] (4.Dx & 4.Ex). The approaches I’ve tried are here: Let ...
Rutvaj Nehete's user avatar
0 votes
0 answers
30 views

What is the full symmetry group of a tile in the shape of a regular n–gon?

I am trying to answer the question on exercise 4.5.6 from the book "Algebra: Abstract and concrete" by Goodman. The chapter is on the symmetries of polyhedra and in this exercise he asks me ...
CoolJedi132's user avatar
0 votes
1 answer
27 views

Algorithm / equations to position a point just outside or inside the edge of a regular polygon?

Here is a polygon with a dot inside an edge, and a dot outside another edge. How do you calculate the $x$ and $y$ position of any dot (whether it's inside or outside of the line's edge) positioned ...
Lance's user avatar
  • 3,773
0 votes
0 answers
28 views

What is the maximum area that can be enclosed in a polygon formed by n wires? [duplicate]

The problem is that we have n wires of different lengths, i.e. $w_1,w_2,...,w_n$. The wires are aligned in a way such that they enclose the maximum area. What is that maximum area, or its best ...
Panda's user avatar
  • 101
0 votes
0 answers
30 views

Necessary and sufficient conditions of the set of interior angles of a polygon

Question Can we find a set of conditions such that for any set $A$ satisfying these conditions, a polygon can be constructed whose set of interior angles is equal to $A$, and for any polygon $P$, the ...
Cristof012's user avatar
1 vote
1 answer
54 views

Maximal irregular polygon inside a regular polygon

Problem: We have a regular $n$-gon. We want to choose some of it's vertices ($A_1, A_2, \ldots, A_m$), so these vertices form a completely irregular $m$-gon. Meaning that all of it's sides have ...
math_inquiry's user avatar
1 vote
1 answer
18 views

Is it always possible to cross two opposite pairs of adjacent sides of a convex 2n-gon by two parallel lines, not crossing through vertex?

We are given a convex polygon with even number of vertexes ($ABCDEF$ in my picture). We want to prove that it is always possible to find two pairs of adjacent sides that have the same amount of ...
Vladimir_U's user avatar
1 vote
0 answers
36 views

Area of Cyclic polygons

Can we generalize /extend Brahmagupta's formula to find the area of cyclic pentagons, hexagons, $n$ sided polygons $n>3$ ? For example $$ 2s= (a+b+c+d+e), \Delta=\sqrt{s (s-a)(s-b)(s-c)(s-d)(s-e)};...
Narasimham's user avatar
  • 41.1k
0 votes
1 answer
39 views

Circular Array - Incoming Wave angle for a Polygon

I am trying to calculate the angle of a plane sound wave arriving at a circular array. A circular array has $M$ receiving elements that are lying on a circle with radius R and the distance between ...
Marco's user avatar
  • 1
2 votes
1 answer
26 views

Number of vertices of product of polytopes

If one takes the product of a polytope $P_1$ with another polytope $P_2$, where $P_1$ has $n$ vertices and $P_2$ has $m$ vertices, will the product of the two polytopes have $mn$ vertices? It is not ...
Tom's user avatar
  • 3,005
0 votes
0 answers
14 views

How do I identify which vertex to start with when constructing an irregular curve of constant width?

A Curve of Constant Width (CoCW) is a closed, convex curve whose width (measured as the distance between parallel supporting lines) is the same in all directions. A CoCW can be constructed from any ...
Lawton's user avatar
  • 1,861
1 vote
1 answer
59 views

Do we lose generality by making this assumption?

Problem : We consider the inscribed pentagon ABCDE in which AB = BC = CD and centroid of the pentagon coincides with the center of the circumscribed circle. Show that pentagon ABCDE is regular. My ...
Unknowduck's user avatar
1 vote
1 answer
55 views

Polygon Problem

Problem: Let $A_1A_2\dots A_{100}$ be a regular $100$-gon with a circumcircle of diameter $1$. Let $M$ be the midpoint of the minor arc $A_1A_2$. Find $$ \sum^{100}_{i=1} |MA_i|^2. $$ My attempt: Let ...
Maximilian Lin's user avatar
5 votes
2 answers
463 views

What's the difference between zig-zags and helixes?

I've been reading through the Polytope-Wiki entry on helices. To my understanding, an $n$-gonal helix is a blend of a planar $n$-gon $\{n\}$ with the regular linear apeirogon $\{\infty\}$. The blend ...
Numeral's user avatar
  • 1,860
2 votes
0 answers
28 views

On the shoelace formulas

I've stumbled on this nice formula to compute the barycenter $\bar{c}$ of an arbitrary (but not self-intersecting) polygon \begin{equation} \bar{c} \triangleq \sum_{i=1}^n \frac{A_i}{A} \bar{c}_i \...
matteogost's user avatar
0 votes
1 answer
77 views

Finding length of sides of a five-sided region within a rectangle when centroid is known

I have a rectangle with a given length $L$ and width $B$. There is a point in one of quadrant of this rectangle whose location $x$ value can vary from $0$ to $B/2$, and $y$ can vary from $0$ to $L/2$. ...
Osama Anwar's user avatar
0 votes
1 answer
57 views

Finding the Areas of Polygons from Side Lengths

I am aware of the formula for the area of a regular polygon: $A=([Side Count] \times [Side Length] \times [Apothem Length])/2$ However, I could not find an equation for the area of a non-regular ...
Don't mail me's user avatar
4 votes
2 answers
192 views

Circumscribed (irregular) pentagon

You're given the consecutive lengths of the sides of an irregular pentagon, and you want to circumscribe this pentagon (whose interior angles are unknown yet) about a circle of unknown radius. Is ...
that's what it is's user avatar
0 votes
1 answer
72 views

Sum of diagonals of pentagon inscribed in circle

Problem I'm trying to solve: Attempt 1: Using the cosine rule we find that $$(IT)^2=14^2+10^2-2(14)(10)\cos(F)\Rightarrow \arccos\bigg(\frac{14^2+10^2-(IT)^2}{2(14)(10)}\bigg)=F$$ Analogously we have ...
d0uble_a_b4ttery's user avatar
0 votes
1 answer
39 views

True or false: In a 2D Poisson process, for every point $P$, there exists a convex $1000$-gon with Poisson points as vertices, that contains only $P$.

I made a Desmos graph that generates $30$ uniformly random black points in a disk, with the centre of the disk in red. I asked myself, "Can I always draw a convex quadrilateral with four of the ...
Dan's user avatar
  • 25.8k
0 votes
1 answer
44 views

Geometry needed to layout points in a grid for a centered hexagonal number?

I asked this as a programming question, but perhaps it is a better math question. What is the math needed to compute the positions of the points along the flat edges of the hexagon? I could then ...
Lance's user avatar
  • 3,773
8 votes
0 answers
135 views

How many sides of an odd-sided equiangular polygon must be proven equal to conclude that it is regular?

If we know that all angles are equal in a polygon with an odd number of sides, how many sides need to be shown to be equal to claim that the polygon is regular? For example, if we know that a triangle ...
Atri De's user avatar
  • 193
6 votes
3 answers
419 views

If all of the angles of a polygon are equal, and half of its sides are equal, then must the polygon be regular? [closed]

If we have that all the angles in a 2d polygon are equal, and if we have that half the sides of the polygon are equal(for example, if the polygon has 10 sides, we know that 5 of them are equal), is it ...
Atri De's user avatar
  • 193
2 votes
0 answers
71 views

A curious connection between the orthocenter of a pentagon and linear algebra

This question is a follow-on of this one which has received a complete answer by @Intelligenti Pauca. Playing around with fig. 1 below, I have found a way to place the issue of the orthocenter of a ...
Jean Marie's user avatar
  • 83.9k
13 votes
4 answers
1k views

The sum of the squares of the diagonals in a polygon

The first question that got me here: A regular dodecagon $P_1 P_2 P_3 \dotsb P_{12}$ is inscribed in a circle with radius $1.$ Compute$ {P_1 P_2}^2 + {P_1 P_3}^2 + {P_1 P_4}^2 + \dots + {P_{10} P_{11}}...
Sai Bhushan's user avatar
0 votes
0 answers
56 views

Tilling of polygons on a sphere

Which convex polygons can be used to tile a sphere? For example, if I had a 20 equilateral triangles, I could form a sphere in the shape of a icosahedron. If I had 2 hexagons for every pentagon, I ...
Anirudh Yamunan Govindarajan's user avatar
1 vote
1 answer
74 views

How to efficiently calculate which region a polygon is in relation to a polynomial with a pencil-and-paper procedure?

A polynomial splits $\mathbb{R}^2$ into 3 regions: above, below, and where the polynomial lies. A polygon is made up of line segments. The polygon's spatial relationship to a polynomial can be one of ...
Teg Louis's user avatar
3 votes
1 answer
106 views

Olympiad Trapezoid Problem about Lengths

I'd like some help with the following Olympiad Problem about a trapezoid: There is a trapezoid $ABCD$ with parallel sides $BC$ and $AD$ such that $AB=1$, $BC=1$, $CD=1$ and $DA=2$. Let $M$ be the ...
CatsAndDogs's user avatar
10 votes
4 answers
1k views

Olympiad Geometry Problem about Pentagon Inscribed in Circle

Someone gave me this question, and I seriously have no clue how to answer it. I've tried considering the centre, drawing diagrams, and searching up all the theorems in the internet but I can't find a ...
CatsAndDogs's user avatar
1 vote
0 answers
71 views

2D tiling with regular pentagons (and generalizations)

We know that the only regular polygons that can tile the 2D plane are triangles, squares, and hexagons. One way of seeing this is that, if we try to place regular pentagons (for instance) around a ...
Rivers McForge's user avatar
0 votes
3 answers
83 views

How to graph an irregular equilateral convex polygon inscribed in a ellipse? [closed]

What would be the steps or equation to graph a irregular convex equilateral polygon inscribed in a ellipse. I have not been able to find a example or equation of this online other then unanswered ...
ItsameLuigi64's user avatar
0 votes
1 answer
19 views

Maximize perimeter among $n$-simplexes with fixed volume

Let $\Omega$ be an $n$-simplex, which means the convex hull of $(n+1)$ points in $\mathbb{R}^n$. We denote $P(\Omega)$ be the perimeter of $\Omega$ and $|\Omega|$ be the volume of $\Omega$. For ...
QFL's user avatar
  • 81
2 votes
1 answer
290 views

Constructing bicentric pentagon

I'm trying to construct a bicentric pentagon in geogebra. I read on Wikipedia that a Pentagon is bicentric if and only if it satisfies this formula $$r(R-x)=(R+x)\left(\sqrt{(R-r)^2-x^2}+\sqrt{2R(R-r-...
PNT's user avatar
  • 4,196
1 vote
0 answers
55 views

Parallelogram Angle Relationship Problem

Question: If one angle of a parallelogram is $24^{\circ}$ less than twice the smallest angle, then what is the measure of the largest angle of the parallelogram? My attempt: First of all, I drew a ...
Utkarsh's user avatar
  • 1,614
0 votes
0 answers
42 views

Notation for referring to specific graph colourings

Consider the following four graphs where $k$ represents the number of colours used to colour the vertices of each graph. Here cycle graphs are used to represent regular polygons, in this specific case ...
Astrid's user avatar
  • 722
1 vote
1 answer
69 views

Getting a point in the interior of a polygon without relying on winding order?

I am given an arbitrary set of points embedded in 3D. The points are guaranteed to be ordered such that their order yields a simple closed polygon, but there is no information about whether they wind ...
Makogan's user avatar
  • 3,439
1 vote
0 answers
62 views

How to find the side-length of a regular polygon given number of sides and diameter [duplicate]

For a program that I am writing, I need a function that takes in the number of sides, and the diameter, and will output side length. I have tried to ask my teachers, but none of them had time to ...
inyourface3445's user avatar
3 votes
1 answer
61 views

Limit of a Sequence of Spiraling Points

The Problem Starting with the vertices $P_1(0,1), P_2(1,1), P_3(1,0), P_4(0,0)$ of a square, we construct further points as shown in the figure: $P_5$ is the midpoint of $P_1P_2$, $P_6$ is the ...
Math Is Fun's user avatar
0 votes
1 answer
100 views

Understanding a proof on IMO shortlist 2016 C3

The problem goes as follow: Let $n$ be a positive integer relatively prime to 6. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour....
H4z3's user avatar
  • 802
-1 votes
1 answer
54 views

Billiard Shot Angles for Circular Table: Return to Starting Point

A circular billiard table is given with a cue ball at the circumference. It is shot at an angle of θ to the line from the ball to the center of the table. For what angles θ will the ball return to ...
Nithin Kamalahasan's user avatar
2 votes
1 answer
84 views

Proof that Polygonal Billiards Have Zero Entropy

I'm reading this paper "Billiards In Polygons" by Boldrighini et al. They say that polygonal billiards have zero measure-theoretic entropy, because a given element of the configuration space ...
interstice's user avatar

1
2 3 4 5
28