Square Quadrants

$\begin{figure}\begin{center}\BoxedEPSF{SquareQuadrants.epsf}\end{center}\end{figure}$

The areas of the regions illustrated above can be found from the equations

$\begin{displaymath} A+4B+4C=1 \end{displaymath}$

(1)

$\begin{displaymath} A+3B+2C={\textstyle{1\over 4}}\pi. \end{displaymath}$

(2)

Since we want to solve for three variables, we need a third equation. This can be taken as

$\begin{displaymath} A+2B+C=2E+D, \end{displaymath}$

(3)

where

$\begin{displaymath} D={\textstyle{1\over 4}}\sqrt{3} \end{displaymath}$

(4)

$\begin{displaymath} D+E={\textstyle{1\over 6}}\pi, \end{displaymath}$

(5)

leading to

$\begin{displaymath} A+2B+C=D+2E=2(D+E)-D={\textstyle{1\over 3}}\pi-{\textstyle{1\over 4}}\sqrt{3}. \end{displaymath}$

(6)

Combining the equations (1), (2), and (6) gives the matrix equation

$\begin{displaymath} \left[{\matrix{1 & 4 & 4\cr 1 & 3 & 2\cr 1 & 2 & 1\cr}}\righ... ...e{1\over 4}}\sqrt{3}\cr}}\right],\hrule width 0pt height 5.9pt \end{displaymath}$

(7)

which can be inverted to yield

$\displaystyle A$	$\textstyle =$	$\displaystyle 1-\sqrt{3}-{\textstyle{1\over 3}}\pi$	(8)
$\displaystyle B$	$\textstyle =$	$\displaystyle -1+{\textstyle{1\over 2}}\sqrt{3}+{\textstyle{1\over 12}}\pi$	(9)
$\displaystyle C$	$\textstyle =$	$\displaystyle 1-{\textstyle{1\over 4}}\sqrt{3}+{\textstyle{1\over 6}}\pi.$	(10)

References

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 67-69, 1991.