Mathematics

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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Mickey Mouse Polygon - Regular Polygons forming Mysterious Angle

The figure shows a regular pentagon and two squares sharing adjacent sides of the pentagon. Some vertices are connected by straight lines as shown. We would like to prove that angle $a$ is exactly $27°$ . The most mysterious thing of the question is…
Chris So
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Regular polygon of radius $1$ with diagonals: mysterious ring of radius $1/e$?

I was playing with a geogebra applet that shows regular $n$ -gons of radius $1$ with their diagonals. For example, here is the $12$ -gon with its diagonals: For any value of $n$ , when I shrink the image, the image becomes darkened (due to the…
Dan
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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…
user23571113
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A regular $n$-gon contains a regular $(n+1)$-gon, with no sides coinciding. What is the maximum number of points of contact between them?

A regular $n$ -gon contains a regular $(n+1)$ -gon. That is, they are in the same plane, and no part of the regular $(n+1)$ -gon is outside of the regular $n$ -gon. None of their sides coincide. There are no other restrictions. Among the integers…
Dan
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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…
Lieuwe Vinkhuijzen
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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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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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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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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 it.…
Arbuja
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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…
Bsonjin
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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 $n$…
Matan
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Simple proof that product of the diagonals of a polygon = N

There is a beautiful fact: If you take a regular N-sided polygon, which is inscribed in the unit circle and find the product of all its diagonals (including two sides) carried out from one corner you will get N exactly: $A_1A_2\cdot A_1A_3\cdot…
klm123
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Is there a question that contains no numbers, whose answer is $\pi/7$?

Is there a question that contains no numbers, except possibly $1$ , whose answer is $\pi/7$ ? There are plenty of questions with no numbers, whose answer is $\pi/n$ for small integer values of $n$ other than $7$ . For some reason, $\pi/7$ seems to…
Dan
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How many lattices does it take to cover a regular $n$-gon?

Given some positive integer $n\ge 3$ , we can ask how many 2-dimensional lattices $L_1,\ldots,L_k$ are required such that their disjoint union contains all vertices of a regular $n$ -gon. (We don't require that the lattices be centered at the…
RavenclawPrefect
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