info prev up next book cdrom email home

Square Quadrants


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

\end{displaymath} (1)

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
\end{displaymath} (3)

D={\textstyle{1\over 4}}\sqrt{3}
\end{displaymath} (4)

D+E={\textstyle{1\over 6}}\pi,
\end{displaymath} (5)

leading to
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
\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)


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

© 1996-9 Eric W. Weisstein