The term "square" can be used to mean either a square number (" is the square of ") or a geometric figure consisting of a convex quadrilateral with sides of equal length that are positioned at right angles to each other as illustrated above. In other words, a square is a regular polygon with four sides.
When used as a symbol, denotes a square geometric figure with given vertices, while is sometimes used to denote a graph product (Clark and Suen 2000).
A square is a special case of an isosceles trapezoid, kite, parallelogram, quadrilateral, rectangle, rhombus, and trapezoid.
The diagonals of a square bisect one another and are perpendicular (illustrated in red in the figure above). In addition, they bisect each pair of opposite angles (illustrated in blue).
The perimeter of a square with side length is
(1)
|
and the area is
(2)
|
The inradius , circumradius , and area can be computed directly from the formulas for a general regular polygon with side length and sides,
(3)
| |||
(4)
| |||
(5)
|
The length of the polygon diagonal of the unit square is , sometimes known as Pythagoras's constant.
The equation
(6)
|
gives a square of circumradius 1, while
(7)
|
gives a square of circumradius .
The area of a square constructed inside a unit square as shown in the above diagram can be found as follows. Label and as shown, then
(8)
|
(9)
|
(10)
|
Expanding
(11)
|
and solving for gives
(12)
|
Plugging in for yields
(13)
|
The area of the shaded square is then
(14)
|
(Detemple and Harold 1996).
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.
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
(15)
|
(Guy 1994). In this problem, one of , , , and 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
(16)
|
In this problem, is divisible by 4 and , , , and are odd. If is not divisible by 3 (5), then two of , , , and are divisible by 3 (5) (Guy 1994).
The centers of four squares erected either internally or externally on the sides of a parallelograms are the vertices of a square (Yaglom 1962, pp. 96-97; Coxeter and Greitzer 1967, p. 84).