Square (geometry)

From Wikipedia, the free encyclopedia

For other uses, see Square.

In plane (Euclidean) geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides. A square is similar to any other square.

The concept of the square is directly related to the square root ( $\sqrt{2}$ ) and to the square exponent (n²).

1 Classification
2 Mensuration formulae
3 Standard coordinates
4 Properties
5 Other facts
6 Non-Euclidean geometry
7 Finite geometry
8 See also
9 External links

[edit] Classification

A square (regular quadrilateral) is a special case of a rectangle as it has four right angles and parallel sides. Likewise it is also a special case of a rhombus, kite, parallelogram, and trapezium.

[edit] Mensuration formulae

The perimeter of a square whose sides have length S is

P = 4 S

And the area is

A = S 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.

[edit] 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 < x_i < 1.

[edit] 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.

If a figure is both a rectangle and a rhombus then it is a square.

[edit] 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 is one of three regular polygons that can form a regular tiling of the plane (the others are the equilateral triangle and the regular hexagon). This is a consequence of the fact that the measure of the angles (90°) is a divisor of 360°.
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 $D 4$ .
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 $\frac{\sqrt{3}}{4}$ units squared, the new area is that of the equilateral triangle with side length S.

[edit] 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.

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.

[edit] Finite geometry

In finite geometry, a subdivided p×p square, with p a prime number, provides a model for a finite geometry with p² points. See finite geometry of the square and cube.