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Square | |||
---|---|---|---|

A square is a regular quadrilateral. | |||

Edges and vertices | 4 | ||

Schläfli symbols | {4} t{2} or {}x{} | ||

Coxeter–Dynkin diagrams | >- | Symmetry group | Dihedral (D_{4}) |

Area (with t=edge length) | t^{2} | ||

Internal angle (degrees) | 90° |

In Euclidean Geometry geometry, a square is a regular polygon with four equal sides. In Euclidean geometry, it has four 90 degree angles. A square with vertices ABCD would be denoted .

The perimeter of a square whose sides have length t is

- $P=4t.$

- $A=t^2.$

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.

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.

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

- It has all equal sides and the angles add up to 360 degrees.
- If a circle is circumscribed around a square, the area of the circle is $pi/2$ (about 1.57) times the area of the square.
- If a circle is inscribed in the square, the area of the circle is $pi/4$ (about 0.79) times the area of the square.
- A square has a larger area than any other quadrilateral with the same perimeter ().
- 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 (in Goldman geometry). There are four lines of reflectional symmetry and it has rotational symmetry through 90°, 180° and 270°. Its symmetry group is the dihedral group $D\_4$.

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.

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.

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}. |

- Square Calculation
- Animated course (Construction, Circumference, Area)
- Definition and properties of a square With interactive applet
- Animated applet illustrating the area of a square

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday October 05, 2008 at 16:48:18 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday October 05, 2008 at 16:48:18 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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