Square in the context of "Rectangle"

In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:$${\displaystyle a^{2}+b^{2}=c^{2}.}$$The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.

In geometry, an octagon (from Ancient Greek ὀκτάγωνον (oktágōnon) 'eight angles') is an eight-sided polygon or 8-gon.

A regular octagon has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, {16}. A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square.

Piazza del Campidoglio ("Capitoline Square") is a public square (piazza) on the top of the ancient Capitoline Hill, between the Roman Forum and the Campus Martius in Rome, Italy. The square includes three main buildings, the Palazzo Senatorio (Senatorial Palace) also known as the Comune di Roma Capitale (City Hall), and the two palaces that make up the Capitoline Museums, the Palazzo dei Conservatori and the Palazzo Nuovo, considered to be one of the oldest national museums, founded in 1471 when Pope Sixtus IV donated some of the museum's most impressive statues, the She-wolf, the Spinario, the Camillus and the colossal head of emperor Constantine. Over the centuries the museums' collection has grown to include many of ancient Roman's finest artworks and artifacts. If something was considered too valuable or fragile in Rome and a copy was made in its place for display, the original is likely now on display in the Capitoline Museum. The hilltop square was designed by Michelangelo in the 16th century. at the behest of Pope Paul III.

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.

In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.

A tessera (plural: tesserae, diminutive tessella) is an individual tile, usually formed in the shape of a square, used in creating a mosaic. It is also known as an abaciscus or abaculus.

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).Two different regions may have the same area (as in squaring the circle); by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

A town square (or public square, urban square, city square or simply square), also called a plaza or piazza, is an open public space commonly found in the heart of a traditional town or city, and which is used for community gatherings. Related concepts are the civic center, the market square and the village green.

Most squares are hardscapes suitable for open markets, concerts, political rallies, and other events that require firm ground. They are not necessarily a true geometric square.