Recent questions in Polygons
ghettoking6q 2022-08-12

### Lebesgue measure using special polygonsI want to prove that the following set satisfies $m\left(G\right)=1$.$G=\left\{\left(x,y\right)\in {\mathbb{R}}^{2}:x>0,0I have to prove this without using the Lebesgue integral.This set is open, so its measure is defined as it follows:$m\left(G\right)=sup\left\{m\left(P\right):P\subset G\right\}$, where P is a special polygon.We can construct a sequence of special polygons ${P}_{n}\subset G$:${P}_{n}=\bigcup _{j=1}^{n}\left[0,log\left(\frac{n+1}{j}\right)\right]×\left[\frac{j-1}{n+1},\frac{j}{n+1}\right].$The measure of one of these special polygons is $m\left({P}_{n}\right)=\sum _{j=1}^{n}log\left(\frac{n+1}{j}\right)\frac{1}{n+1}.$As the sequence of measures $\left\{m\left({P}_{n}\right)\right\}$ is increasing, the supremum of this sequence is its limit:$sup\left(\left\{m\left({P}_{n}\right)\right\}\right)=\underset{n\to \mathrm{\infty }}{lim}\sum _{j=1}^{n}log\left(\frac{n+1}{j}\right)\frac{1}{n+1}=\underset{n\to \mathrm{\infty }}{lim}\mathrm{log}{\left(\frac{\left(n+1{\right)}^{n}}{n!}\right)}^{\frac{1}{n+1}}.$I know that this limit is 1, so the logarithm argument must tend to e, but I don't know how to prove this analytically. I've tried working on the final expression to get the definition of e, but I haven't been able to do so. Could yo help me?

Ashlynn Hale 2022-08-11

### Hyperbolic Angle Measure of PolygonsIs there a way to determine the angle measure of a regular polygon in hyperbolic space? I know that this depends on the length of the sides. A an example, I know that for an equilateral triangle with side length a and angle A, then $\mathrm{sec}A=1+\frac{2{e}^{a}}{{e}^{2a}+1}$.Is there a similar formula for higher regular polygons?

schlichs6d 2022-08-11

### Scan line algorithm for intersecting polygonsGiven two sets of polygons ${P}_{1}=\left\{{s}_{1},...,{s}_{m}\right\}$ and ${P}_{2}=\left\{{s}_{m}+1,...,{s}_{n}\right\}$ with total number of n segments, the previous and next segment on it's polygon can be determined in O(1). Describe a scan-line algorithm that computes all points of ${P}_{1}\cap {P}_{2}$ in O(n).

Marco Hudson 2022-08-11

### Using trigonometric ratios to express area of regular polygonsI am very confused. My book just asked me to use trigonometric ratios to express the area of a regular polygon with 9 sides and lengths of 8.So far I have learned how to use the sin cos and tan in right triangles and have no idea how this applies to all polygons. Can someone please explain this to me

logosdepmpe 2022-08-11

### Constructible polygonsI know certain polygons can be constructed while others cannot. Here is Gauss' Theorem on Constructions:$\mathrm{cos}\left(2\pi /n\right)$ is constructible iff $n={2}^{r}{p}_{1}{p}_{2}···{p}_{k}$, where each ${p}_{i}$ is a Fermat prime.Can this is be used to determine the constructibility of a regular ${p}^{2}$ polygon? If so how and what would be the $\mathrm{cos}\left(2\pi /n\right)$ here?

orkesruim40 2022-08-09

### “Area metric” and “Hausdorff metric” are not equivalent on all closed polygons, but equivalent on convex closed polygonsSuppose X is the set of all closed polygons, ${d}_{\mathrm{\Delta }}$ is the “area metric” defined by the area sum of the symmetric difference of two closed polygons, and dH is the Hausdorff metric on X. How should I prove that ${d}_{\mathrm{\Delta }}$ and ${d}_{H}$ do not generate the same topology on X? Also why do they generate the same topology on the subset of convex polygons? I tried to visualize how a typical open ball in both metrics looks but this seems rather impossible.

Lacey Rojas 2022-08-07

### The area of a triangular sail for a boat is 176 square feel. If the base of the sail is 16 feet long, find its height.

joyoshibb 2022-08-03

### A liquid vessel which is initially empty is in the form of an inverted regular hexagonal pyramid of altitude 25ft and base edge of 10ft. how much will the surface rise when 6,779 liters of water is added?

Luciano Webster 2022-07-25

### Find the lateral surface area of a regular hexagonal pyramid whose edge measures 20 cm and the radius of a circle inscribed in the base is 9 squareroot of 3 cm.

Urijah Estes 2022-07-21

### Can all polygons outside of the largest inscribed rectangle in a convex polygon be concaveLet C be a convex polygon and R the largest (by area) rectangle lying within C. Does there exist a convex polygon C such that when R is removed, all remaining polygons are concave? In other words: Does there exist a polygon C such that all parts of C∖R are concave?

kislotd 2022-07-21