Square

In geometry, a square is a specific type of regular quadrilateral. A quadrilateral is any polygon with four sides. To be a regular quadrilateral, a shape must have equal side lengths and equal interior angles. A square meets both criteria perfectly. It has four straight sides of identical length and four right angles, which measure exactly 90 degrees or $\pi/2$ radians. Because its angles are 90 degrees, the adjacent sides are perpendicular to one another.

A square is a unique shape because it belongs to several different families of polygons. It is a special case of a rectangle, which is defined by having four right angles. It is also a special case of a rhombus, which is defined by having four equal sides. Because it fits both definitions, a square is both a rectangle and a rhombus. This relationship means a square inherits all the properties of these shapes. For example, it is also a parallelogram, a kite, a trapezoid, and a tetragon.

Mathematicians can define a square in many equivalent ways using different geometric properties. You might describe it as a rectangle with four equal sides. Alternatively, you could call it a rhombus that contains at least one right angle. Another way to identify a square is by looking at its diagonals. In a square, the diagonals are equal in length and are perpendicular bisectors of each other. This means they cross at a 90-degree angle and cut each other exactly in half. The diagonals also bisect the internal angles, creating two 45-degree angles at each corner.

Measurement and algebra are closely tied to the properties of the square. The area of a square is calculated by multiplying the side length by itself. This specific operation is so fundamental that mathematicians use the term "squaring" to describe raising any number to the second power.

If a square has a side length of $s$, its perimeter is $4s$. The length of its diagonal is $s\sqrt{2}$. The value of $\sqrt{2}$ is an irrational number, meaning it cannot be expressed as a simple fraction. It is approximately 1.414. This value was known to Babylonian mathematicians as far back as 1800 to 1600 BCE.

Squares possess a high degree of symmetry, more than any other quadrilateral. There are eight rigid transformations that can move a square so that it looks exactly as it did before. These include various rotations around the center and reflections across different axes.

These eight symmetries form a mathematical structure known as the dihedral group of order eight. This group includes combinations of rotations and reflections. When you perform these moves, the square always maps onto itself. This high level of symmetry is why squares are often used as the basic unit in repeating patterns, known as wallpaper groups.

In practical applications, squares are incredibly useful for organization and design. Because equal squares can tile a plane edge-to-edge, they are used for tiled floors, walls, and graph paper. In the digital world, the pixels in bitmap images are often arranged in a square grid.

Even complex data structures, like quadtrees used in image compression, rely on dividing squares into smaller squares. Architecture also frequently uses square footprints. Ancient structures like the Egyptian pyramids and East Asian pagodas feature square bases. Modern skyscrapers often use square plans for practical reasons.

Geometry also explores how squares interact with other shapes. You can fit a circle perfectly inside a square so that it touches the midpoint of each side. This is called an inscribed circle, and its radius is half the side length.

Conversely, a circumscribed circle can be drawn to pass through all four vertices of the square. The square is also the quadrilateral that provides the largest area for a given perimeter. This concept is part of the isoperimetric inequality. Furthermore, while squares are common in Euclidean geometry, they serve as the basis for "metric balls" in non-Euclidean systems like taxicab geometry.