# Questions tagged [polygons]

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 = $[3,1], [3,3], [1,3], [3,5], [7,5], [7,1]$ ...
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### 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 ...
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### 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 ...
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### 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 $0$? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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....
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