For questions on polygons, a flat shape consisting of straight lines that are joined to form a closed chain or circuit
2
votes
0answers
20 views
Determine direction of minimum overlap of convex polygons
Given two convex polygons $P$ and $Q$ what is the minimum intersection polygon $A=P\cap Q'$ where $Q'$ is the polygon $Q$ offset by a vector $\overline r$ of fixed length?
Put another way, what is ...
1
vote
0answers
16 views
Area of a Random Polygon
The following is a long description of a computation I'd like to make. You can think of the process described as a spider randomly building a web. I'd like to know how big we can expect the web to ...
3
votes
1answer
38 views
Symmetrically splitting an octagon into quadrilaterals
I'm wondering whether it is possible to split an octagon into a finite number of quadrilaterals, such that the result is symmetric from all 8 directions (sides or points). There is one condition — any ...
1
vote
1answer
79 views
Is there a pattern in figures whose perimeter is the same as their area?
Here's what I've found so far.
Circumference of a circle with identical circumference and area: $4\pi$
Side length of a triangle with identical perimeter and area: $4\sqrt{3}$
And so on...
...
2
votes
1answer
27 views
Angle Sum of Self-intersecting polygon
If a polygon is self intersecting, is there any way to calculate its angle sum of the interior angles perhaps in terms of the number of sides and the number of points of intersection?
Note we define ...
3
votes
1answer
195 views
cyclic polygons & trigonometry
At one vertex of a pentagon inscribed in a circle of unit diameter (unit diameter, not unit radius) let the angles between adjacent diagonals be $\alpha,\beta,\gamma$, at the next, ...
2
votes
1answer
37 views
Can one use Pick's theorm to prove that area size 5 covers at least 6 grid points?
According to Pick's Theorem, the size of an area $A$ can be calculated by the sum of
the interior lattice points located in the polygon $i$ and the number of lattice points on the boundary placed on ...
1
vote
1answer
26 views
Largest Convex polygon consisting of k points
The problem is Given a set of points, determinate the Largest (in terms of area) Polygon consisting of at most $k$ points.
In a shape like The one below:
$k = 3,polygon =A,F,G $
I would like to ...
1
vote
1answer
37 views
demi dodecahedron - what is it?
What is a demi dodecahedron? I have not been able to find the geometry of a demi dodecahedron.
From latin, what makes a polygon a demi dodecahedron?
Can a demi dodecahedron possibly contain 9 faces? ...
2
votes
2answers
72 views
Number of triangles in a regular polygon
A regular polygon with $n$ sides. Where $(n > 5)$. The number of triangles whose vertices are joining non-adjacent vertices of the polygon is?
6
votes
2answers
63 views
Importance of construction of polygons
Why did the ancient Greeks give so much importance to the construction of regular polygons with $n$-sides using only ruler and compass and tried to study for what $n$ was such a construction possible? ...
1
vote
1answer
42 views
Mathematical notation to describe tiling shapes?
I stumbled across the following Wikipedia article which contained information on tiling by regular polygons.
Underneath each image, it contained a sort of sequence of numbers which appears to be ...
1
vote
2answers
49 views
Sum of all deflection angles.
If a polygon has 42 sides, what should the sum of all the deflection angles be?
I know what a deflection angle is, but I have no clue how to answer this question with the information I've been ...
0
votes
1answer
41 views
I do't understand how to do this problem and was wondering how to get the answer easily 180(n-2) n= the number of sides a figure has.
Please help with this poblem I nedd help with the basics of it.
$180(n-2) n$= the number of sides a figure has.
1
vote
1answer
34 views
Range of Interior Angles of Polygons
$\newcommand{\degree}{{^\circ}}$
Considering the method of of interior angles in surveying a traverse, what is the maximum range of any one interior angle? Also, what is the practical range of any ...
8
votes
2answers
107 views
Why does the term ${\frac{1}{n-1}} {2n-4\choose n-2}$ counts the number of possible triangulations in a polygon?
In the given picture bellow, it counts the number of different triangloations in a polygon, how do the get to this expression, why is it:
$$
{2n-4\choose n-2}
$$
and why do we multiply it by ...
1
vote
3answers
50 views
Are there any Heron-like formulas for convex polygons?
Are there any Heron-like formulas for convex polygons ? By Heron-like I mean formulas without angles as arguments and which takes as arguments only lenghts of sides of polygon - that is - we know no ...
1
vote
0answers
30 views
Position on a path
So the situation is: I have a path (which is represented as a two-dimensional array of GPS coordinates) and I have a percentage position on this path.
I.e. I know that a person has walked 80% of the ...
2
votes
1answer
75 views
How many triangles are formed by $n$ chords of a circle?
This is a homework problem I have to solve, and I think I might be misunderstanding it. I'm translating it from Polish word for word.
$n$ points are placed on a circle, and all the chords whose ...
0
votes
0answers
37 views
Corresponding Triangulations of an (n+2)-gon to n Segments Connecting n+1 Collinear Points
So I'm asked to count the number of ways of connecting n+1 collinear points with n line segments subjected to the following constraints:
If the line is L
1) No segment passes below L.
2) Starting at ...
0
votes
1answer
38 views
Points that are quadrilaterals
When are 4 points in a plane considered quadrilaterals? Also, could you give me examples of 4 points that are not considered quadrilaterals? Thank you in advance!
0
votes
0answers
57 views
Straight skeleton is a tree
Can anybody give me a hint on how to prove that the straight skeleton of every polygon is a tree. Here is the definition of the straight skeleton (taken from Wikipedia):
The straight skeleton of a ...
3
votes
1answer
63 views
Find the vertices of the polytope
Let $x,n$ be 2 integers with $x<n$.
I need to find the vertices of the polytope $P$ of $2 \times n$ nonnegative matrices $A$ such that:
The first row in $A$ is summed to $x$.
$$\sum_{j=1}^n ...
1
vote
4answers
140 views
How can I prove that exterior angles of a pentagon add up to four right angles?
How can I prove that exterior angles of a pentagon add up to four right angles
I have thought about dividing the pentagon into 3 triangles, then maybe using the exterior angle sum equal to two ...
1
vote
1answer
33 views
Find the minimal number of guard points of polygon
Given a polygon with $n$ vertices, what is the minimal number of points inside the polygon such that for each interior point there exists at least one point such that the segment between them lies ...
1
vote
1answer
55 views
How to map points in a unit square to a regular polygon?
I have a set of points in a unit square $x = [-1,1]$ and $y = [-1,1]$ and I want to remap them to their equivalent points in a regular polygon ($n \geq 3$; Triangle and so on).
I've found a really ...
0
votes
2answers
41 views
Property of nonconvex polygons
How to prove that each non-convex polygon with no self-intersecting parts, has at least one interior angle which size is less then $180$ degrees.
2
votes
1answer
149 views
Calculate 'Rectangle' Coordinates Given 2 Points and width
I want to create a rectangular polygon using two points as guides.
So let's say a journey starts in Egypt and ends in London, my polygon should have 4 points:
10 miles further from London ...
2
votes
3answers
137 views
Very confusing polygon question. Can anyone help?
I was practising questions on principles on mathematics. I stumbled onto this question and I don't know where to start. Can anyone please help??
If $P_1P_2....P_n$ is a regular polygon in the ...
3
votes
2answers
48 views
Construct polygon from random segments
Given an arbitrary amount of ordered segments, with arbitrary lengths is there a way to determine if they can be formed into a simple polygon?
And if so, is it possible to work out the angles needed ...
1
vote
1answer
159 views
Non Self Intersecting Polygons?
Given a set of n points is it always possible to construct a non self intersecting polygon?
1
vote
3answers
489 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 ...
0
votes
0answers
80 views
k-dissection of a polygon with non-intersecting diagonals
I am trying to use the vertex coalescing method like the one mentioned here, page 10, to count: Number of dissections of a polygon using non-intersecting diagonals into even number of regions.
I am ...
0
votes
0answers
71 views
Area of a simple random quadrilateral
Given four randomly chosen points with known coordinates, how to compute the area of a not self crossing quadrilateral?
There is a formula if the points are ordered (direct or indirect). So the ...
2
votes
0answers
128 views
Points in the cartesian plane
Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
In the $xy$ plane, suppose $M_{i,j}=\{(x,y)\mid i\le x\le j\}$. ...
3
votes
1answer
127 views
Concave polygons overlapping test
I have set of $N$ concave polygons, given as list of 2D Euclidean coordinates. How to compute:
a. if any of them are overlapping?
b. if one arbitrarily selected polygon overlaps with any of the ...
3
votes
0answers
66 views
Polyonimo Tiling
I came up with the following conjecture the other day, and was wondering if the result was well-known or even true:
Define $f(P)$ for a polyomino $P$ (without holes) to be the least number of total ...
1
vote
1answer
58 views
Diagonal of a convex polygon such that the obtained cuts have simmilar areas
Let $P$ be a convex polygon represented with a list of vertices specified by some orientation. Consider the following problem
Problem. Find in linear time a diagonal of $P$ such that the absolute ...
2
votes
2answers
53 views
What's wrong with an irregular digon?
I recently found that there were some things that could be said about the digon, the polygon with 2 vertices and 2 edges; in particular, the Wikipedia article notes that “in spherical geometry a ...
1
vote
1answer
176 views
Relationship between the sides of inscribed polygons
In my math textbook there's a demonstration for the calculus of the circumference of a circle that involves regular polygons inscribed in the circle, but I don't get it. The book gives the following ...
3
votes
2answers
95 views
Is every triangle a quadrilateral?
I can imagine a quadrilateral where one of the angles is $180^\circ$. Is this still considered a quadrilateral?
More generally, is every $n$-gon also a $(n+1)$-gon (for $n \ge 3$)?
-2
votes
1answer
62 views
Resolving an area of a polygon with sides known
If I have a polygon with the sides given, is it able to calculate the area of it?
If yes, how many sides do I need to know at least?
A formula is appreciated.
I know how to get the area with a ...
1
vote
0answers
83 views
Name and properties of spherical polygon with small-circle sides
Just as the title says: is there a formal name for a convex polygon on a sphere, of which the vertices are connected not by great circle but by small circle segments?
My end goal is to intersect two ...
0
votes
2answers
74 views
Rectangular problem
I was trying to solve this problem:
Let P be a point in the interior of rectangle ABCD. Given PA = 3, PD = 4 and PC = 5, find PB.
I feel lost because it's not right to assume P is in the center ...
2
votes
0answers
105 views
Formal name for polygon with hole
Is there a formal name for an irregular polygon that has 1 or more holes or cutouts in it? I've heard it refered to as a "swiss cheese polygon" or a "Donut polygon". Is this even strictly a polygon?
2
votes
1answer
153 views
Formulas for finding out if a number is Heptagonal or Octagonal
I am trying to find the formula for finding out if a number is heptagonal. I am also looking for the formula for finding out if a number is octagonal.
I already have the formula for finding out the ...
3
votes
1answer
215 views
How can I construct a 2^63-gon with a straightedge and compass?
I entered 2^63 as a stand alone value at WolframAlpha. Among
the responses was a factoid that 'A regular 9223372036854775808-gon is constructible with a straightedge and compass.'
What is such a ...
2
votes
1answer
545 views
How many rectangles can fit in a polygon with n-sides?
I am trying to write an algorithm to solve a problem I have. I have a few ideas of what the algorithm might be like but I am posting to see if anyone else has a better more efficient solution or any ...
