Square

A square and its diagonals

Drawing a square

A square is a shape with four equal sides and four corners that are all right angles (90 degrees). The diagonals of a square also cross at right angles. The angle between any diagonal and a side of a square is 45 degrees. A square has rotational symmetry of four. It has four lines of regular symmetry. A square with vertices A, B, C and D can be written as [math]\displaystyle{ \square ABCD }[/math].^[1]^[2]

A square is a type of rectangle with all sides of equal length. However, while a square is a type of rectangle and rhombus,^[3] the opposite does not need to be true.

A square is also the 2-dimensional analogue of a cube.

Formulas

If the length of side a is known, then we have that:^[2]


Perimeter	[math]\displaystyle{ 4a }[/math]
Area	[math]\displaystyle{ a^2 }[/math]
Length of diagonal	[math]\displaystyle{ \sqrt{2}a }[/math]

If the length of diagonal d is known, then we have that:


Length of side	[math]\displaystyle{ \frac{d}{\sqrt{2}} }[/math]
Perimeter^[4]	[math]\displaystyle{ 2d\sqrt{2} }[/math]
Area^[4]	[math]\displaystyle{ \frac{d^2}{2} }[/math]

Square Media

The area of a square is the product of the length of its sides.
= 2 plotted on Cartesian coordinates.
Square at a given circumcircle
The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the square is r8 and no symmetry is labeled a1.
en:Dihedron {4,2
SVG redraw of Image:Square on sphere.png
Square on hyperbolic plane
Tetrahemihexahedronvertfig
en:Petrie polygon tetrahedron