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The term square is sometimes used to mean Square Number. When used in reference to a geometric figure, however, it means a convex Quadrilateral with four equal sides at Right Angles to each other, illustrated above.

The Perimeter of a square with side length $a$ is

\end{displaymath} (1)

and the Area is
\end{displaymath} (2)

The Inradius $r$, Circumradius $R$, and Area $A$ can be computed directly from the formulas for a general regular Polygon with side length $a$ and $n=4$ sides,
$\displaystyle r$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}a\cot\left({\pi\over 4}\right)={\textstyle{1\over 2}}a$ (3)
$\displaystyle R$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}a\csc\left({\pi\over 4}\right)={\textstyle{1\over 2}}\sqrt{2}\,a$ (4)
$\displaystyle A$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}na^2\cot\left({\pi\over 4}\right)=a^2.$ (5)

The length of the Diagonal of the Unit Square is $\sqrt{2}$, sometimes known as Pythagoras's Constant.

\begin{figure}\begin{center}\BoxedEPSF{SquareDissection.epsf scaled 780}\end{center}\end{figure}

The Area of a square inscribed inside a Unit Square as shown in the above diagram can be found as follows. Label $x$ and $y$ as shown, then

\end{displaymath} (6)

\end{displaymath} (7)

Plugging (6) into (7) gives
\end{displaymath} (8)

\end{displaymath} (9)

and solving for $x$ gives
\end{displaymath} (10)

Plugging in for $y$ yields
\end{displaymath} (11)

The area of the shaded square is then
A=(\sqrt{1+r^2}-x-y)^2={(1-r)^2\over 1+r^2}
\end{displaymath} (12)

(Detemple and Harold 1996).

\begin{figure}\begin{center}\BoxedEPSF{SquareConstruction.epsf scaled 700}\end{center}\end{figure}

The Straightedge and Compass construction of the square is simple. Draw the line $OP_0$ and construct a circle having $OP_0$ as a radius. Then construct the perpendicular $OB$ through $O$. Bisect $P_0OB$ and $P_0'OB$ to locate $P_1$ and $P_2$, where $P_0'$ is opposite $P_0$. Similarly, construct $P_3$ and $P_4$ on the other Semicircle. Connecting $P_1P_2P_3P_4$ then gives a square.

As shown by Schnirelmann, a square can be Inscribed in any closed convex planar curve (Steinhaus 1983). A square can also be Circumscribed about any closed curve (Steinhaus 1983).

An infinity of points in the interior of a square are known whose distances from three of the corners of a square are Rational Numbers. Calling the distances $a$, $b$, and $c$ where $s$ is the side length of the square, these solutions satisfy

\end{displaymath} (13)

(Guy 1994). In this problem, one of $a$, $b$, $c$, and $s$ is Divisible by 3, one by 4, and one by 5. It is not known if there are points having distances from all four corners Rational, but such a solution requires the additional condition
\end{displaymath} (14)

In this problem, $s$ is Divisible by 4 and $a$, $b$, $c$, and $d$ are Odd. If $s$ is not Divisible by 3 (5), then two of $a$, $b$, $c$, and $d$ are Divisible by 3 (5) (Guy 1994).

See also Browkin's Theorem, Dissection, Douglas-Neumann Theorem, Finsler-Hadwiger Theorem, Lozenge, Perfect Square Dissection, Pythagoras's Constant, Pythagorean Square Puzzle, Rectangle, Square Cutting, Square Number, Square Packing, Square Quadrants, Unit Square, von Aubel's Theorem


Detemple, D. and Harold, S. ``A Round-Up of Square Problems.'' Math. Mag. 69, 15-27, 1996.

Dixon, R. Mathographics. New York: Dover, p. 16, 1991.

Eppstein, D. ``Rectilinear Geometry.''

Guy, R. K. ``Rational Distances from the Corners of a Square.'' §D19 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 181-185, 1994.

Steinhaus, H. Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, p. 104, 1983.

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© 1996-9 Eric W. Weisstein