File:FS CQ dia.png

Summary[edit]

Shows the largest square that can be inscribed into a circle.

General case[edit]

Base is the circle around point ${\displaystyle S}$ of radius ${\displaystyle r_{0}}$.

In order to find the side length ${\displaystyle r_{1}}$ of the square ${\displaystyle [ABCD]}$ the following calculations have to be done:

The line line segments ${\displaystyle [AC],[BD]}$ are the diameter of the circle. This leads to
(1)${\displaystyle \quad |AC|=|BD|=2\cdot r_{0}}$

The line segments ${\displaystyle [AD],[AB],[BC],[CB]}$ have the side length ${\displaystyle r_{1}}$ of the square. Thus we have
(2)${\displaystyle \quad |AD|=|AB|=r_{1}}$

Applying the Pythagorean theorem we get:
(3)${\displaystyle \quad |AD|^{2}+|AB|^{2}=|BD|^{2}}$

Using the results of (1) and (2) gives:

${\displaystyle \quad |AD|^{2}+|AB|^{2}=|BD|^{2}}$

${\displaystyle \quad \Leftrightarrow r_{1}^{2}+r_{1}^{2}=(2\cdot r_{0})^{2}}$

${\displaystyle \quad \Leftrightarrow 2\cdot r_{1}^{2}=4\cdot r_{0}^{2}}$

${\displaystyle \quad \Leftrightarrow r_{1}^{2}=2\cdot r_{0}^{2}}$

${\displaystyle \quad \Leftrightarrow r_{1}={\sqrt {2}}\cdot r_{0}}$

Segments in the general case[edit]

0) The radius of the circle ${\displaystyle =r_{0}}$
1) The side length of the inscribed square ${\displaystyle r_{1}={\sqrt {2}}\cdot r_{0}}$

Perimeters in the general case[edit]

0) Perimeter of base circle ${\displaystyle P_{0}=2\pi \cdot r_{0}}$
1) Perimeter of the square ${\displaystyle P_{1}=4\cdot r_{1}=4\cdot {\sqrt {2}}\cdot r_{0}}$

Areas in the general case[edit]

0) Area of the base circle ${\displaystyle A_{0}=\pi r_{0}^{2}}$
1) Area of the square ${\displaystyle A_{1}=r_{1}^{2}=({\sqrt {2}}\cdot r_{0})^{2}=2\cdot r_{0}^{2}}$

Covered surface of the base shape: ${\displaystyle C_{b}={\frac {A_{1}}{A_{0}}}={\frac {2\cdot r_{0}^{2}}{\pi \cdot r_{0}^{2}}}={\frac {2}{\pi }}\quad \approx 63.7\%}$

Centroids in the general case[edit]

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

Normalised case[edit]

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

Segments in the normalised case[edit]

0) Radius of the base circle ${\displaystyle r_{0}={\frac {1}{\sqrt {\pi }}}=0.5641896...}$
1) Side length of the square ${\displaystyle r_{1}={\sqrt {2}}\cdot {\frac {1}{\sqrt {\pi }}}={\sqrt {\frac {2}{\pi }}}=0.7978846...}$

Perimeters in the normalised case[edit]

0) Perimeter of base circle ${\displaystyle P_{0}=2\pi \cdot {\frac {1}{\sqrt {\pi }}}=2{\sqrt {\pi }}=3.5449077...}$
1) Perimeter of the square ${\displaystyle P_{1}=4\cdot {\sqrt {\frac {2}{\pi }}}=3.1915382...}$
S) Sum of perimeters ${\displaystyle P_{s}=P_{0}+P_{1}=6.7364459...}$

Areas in the normalised case[edit]

0) Area of the base circle ${\displaystyle A_{0}=1}$
1) Area of the square ${\displaystyle A_{1}=\left({\sqrt {\frac {2}{\pi }}}\right)^{2}={\frac {2}{\pi }}=0.6366198...}$

Centroids in the normalised case[edit]

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

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