File:FS QC dia.png

English: QC 1.1415927 dia - Largest circle in a square

Shows the largest circle within a square.

General case

Base is the square ${\displaystyle [ABCD]}$ of side length ${\displaystyle a_{0}}$.
The radius ${\displaystyle r_{1}}$ of the inscribed circle has the length ${\displaystyle r_{1}={\frac {a_{0}}{2}}}$

Segments in the general case

0) Side length of the base square: ${\displaystyle a_{0}}$
1) Radius of the inscribed circle: ${\displaystyle r_{1}={\frac {a_{0}}{2}}}$

Perimeters in the general case

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

Areas in the general case

0) Area of the base square ${\displaystyle A_{0}=a_{0}^{2}}$
1) Area of the inscribed circle ${\displaystyle A_{1}=\pi \cdot r_{1}^{2}=\pi \cdot ({\frac {a_{0}}{2}})^{2}=\pi \cdot {\frac {a_{0}^{2}}{4}}={\frac {\pi }{4}}\cdot a_{0}^{2}={\frac {\pi }{4}}\cdot A_{0}}$

Centroids in the general case

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

Normalised case

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

Segments in the normalised case

0) Side length of the base square ${\displaystyle a_{0}=1}$
1) Radius of the inscribed circle ${\displaystyle r_{1}={\frac {1}{2}}=0.5}$

Perimeters in the normalised case

0) Perimeter of base square ${\displaystyle P_{0}=4}$
1) Perimeter of the inscribed circle ${\displaystyle P_{1}=2\cdot \pi \cdot r_{1}=2\cdot \pi \cdot {\frac {1}{2}}=\pi =3.1415926...}$

Areas in the normalised case

0) Area of the base square ${\displaystyle A_{0}=1}$
1) Area of the inscribed circle ${\displaystyle A_{1}=\pi \cdot r_{1}^{2}=\pi \cdot ({\frac {1}{2}})^{2}={\frac {\pi }{4}}=0.785398...}$

Centroids in the normalised case

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

Identifying number

Apart of the base element there is only one other shape allocated. Therefore the integer part of the identifying number is 1.
The decimal part of the identifying number is the decimal part of the sum of the perimeters and the distances of the centroids in the normalised case.

${\displaystyle decimalpart(3.1415927...+0)=decimalpart(3.1415927...)=0.1415927...}$

So the identifying number is: ${\displaystyle 1.1415927}$