Square (geometry)

Definition

A square is a regular quadrilateral, which means that it has four congruent sides and four congruent angles (90-degree angles, or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted as square ABCD.

Properties

A square has several interesting properties, some of which are also possessed by other quadrilaterals.

Sides

All four sides of a square are of equal length. If the length of a side of the square is 'a', then the perimeter of the square is 4a.

Angles

All four angles of a square are equal (each being 90 degrees, a right angle).

Diagonals

The diagonals of a square are equal and bisect each other at right angles. The diagonals of a square also bisect its angles. The length of the diagonal 'd' of a square can be calculated using the formula: d = a√2, where 'a' is the length of a side of the square.

Area

The area 'A' of a square can be calculated using the formula: A = a^2, where 'a' is the length of a side of the square.

In other geometries

Squares exist not only in Euclidean geometry, but also in other geometries.

Taxicab geometry

In Taxicab geometry, the square remains a regular polygon, but it looks different from the square in Euclidean geometry. It has four right angles, but its sides are not all the same length. The square in taxicab geometry is a rotated square in Euclidean geometry.

Spherical and hyperbolic geometry

In Spherical geometry and Hyperbolic geometry, squares do not exist. However, there are figures that are analogous to squares in these geometries.

Squares in algebra

Squares are fundamental in algebra, with the operation of squaring being key to the definitions of many algebraic structures, including fields and rings.

Squares in number theory

In Number theory, the concept of a square is further generalized to the concept of a square number. A square number is an integer that is the square of another integer.

Squares in art and culture

Squares have a rich history in art and culture. They are often used as a symbol of stability, order, and reliability.