Questions tagged [polygons]
For questions on polygons, a flat shape consisting of straight lines that are joined to form a closed chain or circuit
1,451 questions
0 votes
1 answer
51 views
Why aren't there infinite star polytopes?
Surely we can assemble, from some schlafli symbol $\{p/q,n/m\}$ some arbitrary regular polytope? I understand that some of these constructions are infeasible, but surely not all of them are? Could we ...
18 votes
3 answers
407 views
Why does every rose curve contain a regular polygon?
I was playing around in Desmos looking at rose-shaped curves, a family of curves with polar equation $$ r = \cos n \theta, \ \ \ \ \ n \in \mathbb{N}. $$ The number of petals on this rose-curve is ...
- 578
0 votes
0 answers
30 views
Point that induces half-spaces that cut closed polygon into parts with at least a third of the area
Here is the problem. It is a homework problem and I am not sure how to proceed. We have a closed convex polygon $M \subset R^2$. Let its area be $A(M)$. Prove that there is a point $x$ such that every ...
- 55
0 votes
1 answer
60 views
I need help to understand the algorithm that tests if a point lies inside or outside a general 2D polygon
The algorithm in question is from this webpage. The complete algorithm from this webpage is as follows: ...
- 101
0 votes
2 answers
82 views
Finding the interior and exterior angles of a regular nonagon
Quadrilateral angle sum: I know that this would have to equal 360 degrees. $\angle A=90, \angle C=140$ so the angle inside the quadrilateral is $360-140=220$ but I'm not sure how to get $\angle ABC$ ...
- 2,134
7 votes
2 answers
447 views
Estimating the area of a polygon given its perimeter and maximum distance between two vertices.
Let $s$ be the semi-perimeter of a convex polygon of $n$-sides and let $d$ denote the maximum distance between any two vertices of the polygon (i.e., the polygon's diameter). Given this information, ...
- 24.8k
8 votes
0 answers
348 views
Showing that a $1:\sqrt{2}:2$ triangle can be inscribed in a regular heptagon in this way
The special right triangle with sides $1,\sqrt{3},2$ is well known. But consider the triangle with sides $1,\sqrt{2},2$: Here $AB=2,BC=\sqrt{2},CA=1$. This seems uninteresting at first, but it turns ...
- 1
3 votes
0 answers
72 views
Investigating the Locus of the Centroid of a Regular m-gon Rolling Inside a Regular n-gon
I started by solving a problem where a square with side length equal to that of a regular octagon rolls in one direction inside the octagon, and I calculated both the path and the distance traveled by ...
- 31
0 votes
2 answers
250 views
Conjecture on the existence of convex and concave polygons with all prime interior angles
Recently, I have been thinking about an interesting conjecture which I believe has never been preposed before, which I have dubbed the panprimangular polygon conjecture. The conjecture states: $\...
0 votes
1 answer
144 views
Problem with proof in Spivak's Calculus
This is question (b) from problem 11 in Chapter 8 and it has to do with approximating the area of the circle by inscribing appropriate polygons. If $P$ is a regular polygon inscribed inside a circle ...
- 437
53 votes
18 answers
5k views
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 $...
- 681
0 votes
0 answers
90 views
What pattern do area decomposition identities on chords follow? Examples for subtended angles 60°, 72°, 90°, 120°
While playing with circle intersections and their common chords, I noticed that in certain symmetric configurations it is possible to find circle–inscribed area decompositions that resemble the square ...
- 317
1 vote
0 answers
45 views
Given a polygon P and a set of polygons S find the minimum number of polygons in S that covers P.
The union of the polygons you choose don't necessarily need to be equal exactly to the polygon P. I just need the minimum number of polygons in S that cover all of P so we may cover also a part beside ...
- 11
8 votes
3 answers
257 views
Counting triangles in a regular polygon where two angles differ by $90^\circ$
I have recently been studying triangles in which the difference between two of their angles equals $90°$, and I have discovered several interesting properties of such triangles. Now I am interested ...
- 4,461
1 vote
2 answers
286 views
Excircles on polygons with more than 4 sides
Is it possible for a polygon with more than 4 sides to have at least one excircle? The Wikipedia articles I've found on them only mention triangles and quadrilaterals: https://en.wiktionary.org/wiki/...
- 313