Square

For other uses, see Square (disambiguation).

In geometry, a square is a regularquadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles).^[1] It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ${\displaystyle \square }$ ABCD.

Properties

A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a parallelogram (all opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and a rectangle (opposite sides equal, right-angles) and therefore has all the properties of all these shapes, namely:^[5]

• The diagonals of a square bisect each other and meet at 90°
• The diagonals of a square bisect its angles.
• Opposite sides of a square are both parallel and equal in length.
• All four angles of a square are equal. (Each is 360°/4 = 90°, so every angle of a square is a right angle.)
• All four sides of a square are equal.
• The diagonals of a square are equal.
• The square is the n=2 case of the families of n-hypercubes and n-orthoplexes.
• A square has Schläfli symbol {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}.

Perimeter and area

The area of a square is the product of the length of its sides.

The perimeter of a square whose four sides have length ${\displaystyle \ell }$ is

${\displaystyle P=4\ell }$

and the area A is

${\displaystyle A=\ell ^{2}.}$

Square
A regular quadrilateral
Type	Regular polygon
Edges and vertices	4
Schläfli symbol	{4}
Coxeter diagram
Symmetry group	Dihedral (D4), order 2×4
Internal angle (degrees)	90°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal