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Encyclopedia > Square (geometry)
Square

A square
The sides of a square and its diagonals meet at right angles.
Edges and vertices 4
Schläfli symbols {4}
{}x{}
Coxeter–Dynkin diagrams
Symmetry group Dihedral (D4)
Area
(with t=edge length)
t2
Internal angle
(degrees)
90°

## Mensuration formulae

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

The perimeter of a square whose sides have length t is Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... The perimeter is the distance around a given two-dimensional object. ...

P = 4t.

A = t2.

In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term square to mean raising to the second power.

## Standard coordinates

The coordinates for the vertices of a square centered at the origin and with side length 2 are (±1, ±1), while the interior of the same consists of all points (x0, x1) with −1 < xi < 1.

## Properties

Each angle in a square is equal to 90 degrees, or a right angle.

The diagonals of a square are equal. Conversely, if the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are $sqrt{2}$ (about 1.41) times the length of a side of the square. This value, known as Pythagoras’ constant, was the first number proven to be irrational. A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ... Two rhombi. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ...

If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths) then it is a square.

## Other facts

• If a circle is circumscribed around a square, the area of the circle is π / 2 (about 1.57) times the area of the square.
• If a circle is inscribed in the square, the area of the circle is π / 4 (about 0.79) times the area of the square.
• A square has a larger area than any other quadrilateral with the same perimeter ([1]).
• A square tiling is one of three regular tilings of the plane (the others are the equilateral triangle and the regular hexagon).
• The square is in two families of polytopes in two dimensions: hypercube and the cross polytope. The Schläfli symbol for the square is {4}.
• The square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry through 90°, 180° and 270°. Its symmetry group is the dihedral group D4.
• If the area of a given square with side length S is multiplied by the area of a "unit triangle" (an equilateral triangle with side length of 1 unit), which is $tfrac{sqrt{3}}{4}$ units squared, the new area is that of the equilateral triangle with side length S.

In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... Plane tilings by regular polygons have been widely used since antiquity. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... For other uses, see Hexagon (disambiguation). ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... In mathematics, the SchlÃ¤fli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Figures with the axes of symmetry drawn in. ... The triskelion appearing on the Isle of Man flag. ... The symmetry group of an object (e. ... This article may be confusing for some readers, and should be edited to enhance clarity. ...

## Non-Euclidean geometry

In non-euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles.

In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle. Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...

In hyperbolic geometry, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger squares have smaller angles. Lines through a given point P and asymptotic to line l. ...

Examples:

 Six squares can tile the sphere with 3 squares around each vertex and 120 degree internal angles. This is called a spherical cube. The Schläfli symbol is {4,3}. Squares can tile the Euclidean plane with 4 around each vertex, with each square having an internal angle of 90 degrees. The Schläfli symbol is {4,4}. Squares can tile the hyperbolic plane with 5 around each vertex, with each square having 72 degree internal angles. The Schläfli symbol is {4,5}.

Image File history File links Size of this preview: 600 Ã— 599 pixelsFull resolution (816 Ã— 815 pixel, file size: 254 KB, MIME type: image/png) Example square (geometry) on sphere with 120 deg internal angles. ... External angles law In geometry, an interior angle (or internal angle) is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon. ... In mathematics, the SchlÃ¤fli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Image File history File links Size of this preview: 590 Ã— 600 pixelsFull resolution (608 Ã— 618 pixel, file size: 57 KB, MIME type: image/png) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Square from Square tiling I, the copyright holder of this work, hereby release it into the public... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, the SchlÃ¤fli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Image File history File links Size of this preview: 589 Ã— 600 pixelsFull resolution (613 Ã— 624 pixel, file size: 74 KB, MIME type: image/png) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Square from Order-5 square tiling I, the copyright holder of this work, hereby release it into... In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. ... In mathematics, the SchlÃ¤fli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...

In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... Upright square tiling. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... The unit square in a Cartesian coordinate system with coordinates (x,y) is defined as the square consisting of the points where both x and y lie in the unit interval from 0 to 1. ...

Results from FactBites:

 Square (geometry) Summary (550 words) Squares are a special case of regular quadrilateral, rectangle, rhombus, kite, parallelogram, and isosceles trapezoid/trapezium. The Schläfli symbol for the square is {4}. In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles.
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