File:FS CS dia.png

Summary[edit]

Displays the largest equilateral hexagon within a circle

Elements[edit]

Base is the circle of given radius ${\displaystyle r_{0}}$ around point ${\displaystyle S_{0}}$
Inscribed is the largest possible equilateral triangle.

General case[edit]

Segments in the general case[edit]

0) The radius of the base circle: ${\displaystyle r_{0}=|AS_{0}|=|BS_{0}|}$
1) Side length of the hexagon: ${\displaystyle a_{1}=|AB|=|AS_{0}|=r_{0}\quad }$, since the hexogon is equilateral

Perimeters in the general case[edit]

0) Perimeter of base circle ${\displaystyle P_{0}=2\cdot \pi \cdot r_{0}}$
1) Perimeter of the hexagon ${\displaystyle P_{1}=6\cdot a_{1}=6\cdot r_{0}}$
S) Sum of perimeters: ${\displaystyle P_{s}=P_{0}+P_{1}=(2\pi +6)\cdot r_{0}}$

Areas in the general case[edit]

0) Area of the base circle ${\displaystyle A_{0}=\pi \cdot r_{0}^{2}}$
1) Area of the inscribed hexagon ${\displaystyle A_{1}={\frac {3{\sqrt {3}}}{2}}\cdot r_{0}^{2}\quad }$

Covered surface of base shape: ${\displaystyle C_{b}={\frac {A_{1}}{A_{0}}}={\frac {3{\sqrt {3}}}{2\pi }}\quad \approx 82.7\%}$

Centroids in the general case[edit]

Centroid positions are measured from the centroid point of the base shape
0) Centroid positions of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid positions of the inscribed hexagon: ${\displaystyle S_{1}=S_{0}=0+0i}$

Normalised case[edit]

In the normalised case the area of the base is set to 1.
${\displaystyle A_{0}=1\quad \Rightarrow \pi \cdot r_{0}^{2}=1\quad \Rightarrow r_{0}^{2}={\frac {1}{\pi }}\quad \Rightarrow r_{0}={\frac {1}{\sqrt {\pi }}}}$

Segments in the normalised case[edit]

0) Radius of the base circle ${\displaystyle r_{0}={\frac {1}{\sqrt {\pi }}}\quad \approx 0.564}$
1) Side length of the inscribed hexagon ${\displaystyle a_{1}=r_{0}={\frac {1}{\sqrt {\pi }}}\quad \approx 0.564}$

Perimeters in the normalised case[edit]

0) Perimeter of base square ${\displaystyle P_{0}=2\cdot \pi \cdot {\frac {1}{\sqrt {\pi }}}=2\cdot {\sqrt {\pi }}\quad \approx 3.5449077}$
1) Perimeter of the inscribed hexagon ${\displaystyle P_{1}=6\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx 3.3851375}$
S) Sum of perimeters ${\displaystyle P_{s}=P_{0}+P_{1}=(2\pi +6)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx 6.9300452}$

Areas in the normalised case[edit]

0) Area of the base square ${\displaystyle A_{0}=1}$
1) Area of the inscribed hexagon ${\displaystyle A_{1}={\frac {3{\sqrt {3}}}{2}}\cdot \left({\frac {1}{\sqrt {\pi }}}\right)^{2}={\frac {3{\sqrt {3}}}{2\pi }}\quad \approx 0.827}$

Centroids in the normalised case[edit]

Centroid positions are measured from the centroid point of the base shape.
0) Centroid positions of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid positions of the inscribed hexagon:${\displaystyle S_{1}=0+0i}$

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue