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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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 ...
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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 ...
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2answers
31 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 ...
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3answers
41 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $A_1$ ...
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3answers
110 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: ...
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2answers
76 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 ...
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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 ...
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0answers
97 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 ...
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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. ...
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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 ...
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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 ...
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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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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 ...
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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), ...
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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 ...
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1answer
43 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: $17$ ...
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1answer
24 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 ...
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2answers
53 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 ...
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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.
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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 ...
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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 ...
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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 ...
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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} ...
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2answers
32 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 ...
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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 + ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
21 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 ...
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1answer
35 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 ...
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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 ...
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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 ...
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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 ...
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0answers
49 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 ...
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1answer
66 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 ...
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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 ...
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2answers
39 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 ...
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1answer
52 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 ...
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3answers
36 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 ...
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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 ...
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3answers
69 views

Number of sides of regular polygon

What I need to find: Number of sides of a regular polygon What I am given: Any 3 vertices of the polygon What I currently know: I can find the center of the polygon. That would be the intersection ...
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1answer
37 views

Does the inner pentagon inside a Robbins pentagon $also$ have a rational area?

The Heron triangle has integer sides and area. The Robbins pentagon is just the generalization: it also has integer sides and area. The example below has sides $78, 126, 66, 50, 32$ and area $A_R = ...
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24 views

polygon angle names

Regular polygons can be divided up into triangles of equal sizes. For example, a pentagon would have 5 triangles. Each would have one angle with a value of 72 degrees and two angles of 54 degrees. ...