Questions tagged [polygons]

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

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain or circuit.

A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. An $n$-gon is a polygon with $n$ sides. The interior of the polygon is sometimes called its body. A polygon is a $2$-dimensional example of the more general polytope in any number of dimensions.

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If I stretch a convex polygon, does the original fit into the streched version?

Suppose you have a convex polygon $P=\mathrm{conv}(\{(x_1,y_1),\dots, (x_k,y_k)\})$ and you stretch it in one dimension, that is, we choose $\alpha>1$ and get a new polygon $P^\alpha=\mathrm{conv}(\{(\alpha x_1,y_1),\dots, (\alpha x_k,y_k)\})$. Is…
Dart
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Prove that for one vertex of a convex pentagon, the sum of distances to the other four is greater than the perimeter

The problem is from the journal 'Crux Mathematicorum', originally proposed by Paul Erdős and Esther Szekeres for the case of a convex $n$-gon with $n > 5$, and can be found here together with a proof for that case (pdf page 20) . Unfortunately they…
user23571113
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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 whose area is $S$, then find the max of…
mathlove
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Does a random set of points in the plane contain a large empty convex polygon?

Suppose I choose $n$ points uniformly at random from the unit square $[0,1]\times [0,1]$, obtaining a set of points $S=\{p_1,\ldots, p_n\}\subset [0,1]\times [0,1]$. Then $S$ may contain subsets which span an empty convex polygon. For example, in…
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Do side-rational triangles of the same area admit side-rational dissections?

Call a polygon side-rational if the lengths of all its sides are rational. Call a dissection of a polygon side-rational if all of the polygons within the dissection are side-rational. Then my question is as in the title: Do any two rational…
Steven Stadnicki
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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 isolated. Intersection of the lines outside the $n$-gon…
Matan
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If a regular polygon has a fixed edge length, can I know how many edges it has by knowing the length from corner to its center?

So I wonder if there is a formula so that when there's a defined edge length, I can calculate a regular polygon's edges amount by knowing its length from corner to center, or vice versa. So let's say our defined edge length is 1, then by knowing…
Andrew.Wolphoe
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How many triangles can be formed by the vertices of a regular polygon of $n$ sides?

How many triangles can be formed by the vertices of a regular polygon of $n$ sides? And how many if no side of the polygon is to be a side of any triangle ? I have no idea where I should start to think. Can anyone give me some insight ? Use…
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Polygons with equal area and perimeter but different number of sides?

Let's say we have two polygons with different numbers of sides. They can be any sort of shape, but they have to have the same area, and perimeter. There could be such possibilities, but can someone show me with pictures? I just need visualize…
Arbuja
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Convex polygons that do not tile the plane individually, but together they do

I am looking for two convex polygons $P,Q \subset \Bbb R^2$ such that $P$ does not tile the plane, $Q$ does not tile the plane, but if we allowed to use $P,Q$ together, then we can tile the plane. Here I do not require the tilings to be lattice…
Alphonse
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Can a parallelogram have whole-number lengths for all four sides and both diagonals?

Is it possible for a parallelogram to have whole-number lengths for all four sides and both diagonals? One idea I had was to arrange four identical right triangles such that the right angles are adjacent. For example if we take four triangles that…
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Polygon-in-polygon testing

I have a list of vertices of simple polygons, and I would like to test whether or not a polygon is fully contained in another polygon in the list. Is it sufficient to do something like: Let $p_0$ be the candidate polygon. Let $r_i, ~le_i, ~u_i,…
WeakLearner
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proving that the area of a 2016 sided polygon is an even integer

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90$°or $270$°. If the lengths of its sides are odd integers, prove that its area is an even integer. I think…
space
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Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex configuration, it is always possible to shift a point in…
user139000
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Area of the Limiting Polygon

Start with an equilateral triangle with unit area. Trisect each of the sides and then cut-off the corners. In this case, we get a regular hexagon - see the picture below. Next, trisect each of the sides of the hexagon and cut-off the corners. This…
Fly by Night
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