| 释义 |
SquareThe term square is sometimes used to mean Square Number. When used in reference to a geometric figure, however, itmeans 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) |
 , Circumradius  , and Area  can be computed directly from the formulas for ageneral regular Polygon with side length and  sides,
The length of the Diagonal of the Unit Square is  , sometimes known asPythagoras'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 circlehaving 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 canalso 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 areRational Numbers. Calling the distances ,  , and  where  is the side length of thesquare, these solutions satisfy
 | (13) |
 | (14) |
 are Odd. If is not Divisible by 3 (5),then two of , , , and 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, UnitSquare, 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.
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