Square

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

$$L=4a$$ (1)

and the Area is

$$A=a^2.$$ (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,

$$r = {\textstyle{1\over 2}}a\cot\left({\pi\over 4}\right)={\textstyle{1\over 2}}a$$ (3)

$$R = {\textstyle{1\over 2}}a\csc\left({\pi\over 4}\right)={\textstyle{1\over 2}}\sqrt{2}\,a$$ (4)

$$A = {\textstyle{1\over 4}}na^2\cot\left({\pi\over 4}\right)=a^2.$$ (5)

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

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

$$x^2+y^2=r^2$$ (6)

$$(\sqrt{1+r^2}-x)^2+y^2=1.$$ (7)

Plugging (6) into (7) gives

$$(\sqrt{1+r^2}-x)^2+(r^2-x^2)=1.$$ (8)

Expanding

$$x^2-2x\sqrt{1+r^2}+1+r^2+r^2-x^2=1$$ (9)

and solving for $x$ gives

$$x={r^2\over\sqrt{1+r^2}}.$$ (10)

Plugging in for $y$ yields

$$y=\sqrt{r^2-x^2}={r\over\sqrt{1+r^2}}.$$ (11)

The area of the shaded square is then

$$A=(\sqrt{1+r^2}-x-y)^2={(1-r)^2\over 1+r^2}$$ (12)

(Detemple and Harold 1996).

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

$$(s^2+b^2-a^2)^2+(s^2+b^2-c^2)^2=(2bs)^2$$ (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

$$a^2+c^2=b^2+d^2.$$ (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

References

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.'' http://www.ics.uci.edu/~eppstein/junkyard/rect.html.

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.