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. 9697; Coxeter and Greitzer 1967, p. 84).