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 is

(1) |

(2) |

(3) | |||

(4) | |||

(5) |

The length of the Diagonal of the Unit Square is , 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
and as shown, then

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

The Straightedge and Compass construction of the square is simple. Draw the line and construct a circle having as a radius. Then construct the perpendicular through . Bisect and to locate and , where is opposite . Similarly, construct and on the other Semicircle. Connecting 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 , , and where is the side length of the
square, these solutions satisfy

(13) |

(14) |

**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.

© 1996-9

1999-05-26