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

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2
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1answer
496 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) ...
6
votes
1answer
65 views
+50

can a convex polygon have only one boundary point at locally maximum distance from its centroid?

It's easy to see that given any convex polygon P and any point c in its interior, there is at least one point m on the boundary of P at locally maximum distance from c: simply choose m to be a vertex ...
0
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1answer
383 views

Minkowski difference of two convex polygons

I just want to make sure that the following algorithm is correct for computing the Minkowski difference of two shapes $A,B$: $\text{Minkowski}(A,B) = \text{ CH } \{x: x = a - b \text{ for } a \in ...
9
votes
1answer
51 views

How do you calculate the smallest cycle possible for a given tile shape?

If you connect together a bunch of regular hexagons, they neatly fit together (as everyone knows), each tile having six neighbors. Making a graph of the connectivity of these tiles, you can see that ...
0
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0answers
20 views

What is the name of a “polygon” with piece-wise polynomial boundary?

I would like to know if somebody knows the name of these objects. Given a set of $N$ vertices $\{(x_i, y_i)\}_{i=1,\ldots,N}$ (points in $\mathbb{R}^2$) we create a closed curve, defined piece-wise, ...
0
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0answers
15 views

Using GPS coordinates in trillateration

for a project we need to find a certain position. The info we have : 3 surrounding positions and the distance between those positions and the point we are looking for. We've got a setup like this ...
9
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3answers
4k views

How is the area of a country calculated?

As countries' or states' borders are not straight lines but they are irregular in nature, I wonder how anyone can calculate the area of a country or state. When do you think the area of a country or ...
-4
votes
0answers
50 views

Isoperimetric hexagon and triangle ; comparing their areas. [on hold]

A regular hexagon ( all sides of equal length and all angles equal ) and an equilateral triangle is equal circumference. What percent larger is the largest area ??
10
votes
2answers
2k views

Can a polygon be one dimensional?

When looking up the definition of polygon, Wikipedia tells me: In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing ...
0
votes
2answers
631 views

Can a polygon have four 90 degree corners and still not be a rectangle?

On another woodworking forum, someone said that after building a case, you should measure the diagonals to ensure the case is square and that just checking if all the corners are 90 degrees won't ...
1
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0answers
12 views

What's the difference between the Minkowski difference of $A$ and $B$ and the Minkowski sum of $A$ and $-B$?

In the book Computational Geometry, Algorithms and Applications from de Berg, van Kreveld, Overmars and schwarzkopf, I read the following in chapter 13.3 on Minkowski sums: Sometimes $ P ...
1
vote
1answer
27 views

The sum of distances from the sides of a regular polygon to an interior point is a constant

Let there be a regular polygon of $n$ sides. Assume there is a point $P$ inside the polygon, then prove that $$a_1 + a_2 + a_3 + \cdots + a_n= \text{constant}$$ where $a_i$ is the distance of ...
2
votes
2answers
32 views

Can a polygon with an infinite number of sides be viewed as a line?

The inner angles of a polygon approach 180º as the number of sides (N) of the polygon increases. So, if N approaches infinity, we would have a circle. But... At infinity, we would also have a set of ...
2
votes
1answer
51 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
3answers
46 views

A simple proof that a polygon circumscribing a circle overestimates its perimeter

Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side ...
2
votes
0answers
36 views

Number of quadrilaterals formed out of N points.

How can I calculate the number of $4$-vertex convex polygons (quadrilaterals) that can be formed out of $n$ given points, where $n \geq 4$? Note: Points can be collinear. So triangles with $3$ sides ...
19
votes
7answers
814 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 ...
16
votes
2answers
418 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 ...
2
votes
0answers
16 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
48 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 $A_1$ ...
10
votes
3answers
111 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
19 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
77 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
44 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
99 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 ...
1
vote
1answer
78 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
41 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
23 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 ...
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0answers
19 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
372 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
48 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
63 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), ...
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
54 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
25 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
62 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
1answer
24 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 ...
1
vote
1answer
55 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
12 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
46 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
86 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 ...
0
votes
2answers
33 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
39 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
74 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
64 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
37 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
77 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 ...