Square

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).

Classification

A square (regular quadrilateral) is a special case of a rectangle, rhombus, kite, and parallelogram.

Mensuration formulae

The perimeter of a square whose sides have length S is

$P=4S$

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.

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.

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.

A square is the only quadrilateral that can be every quadrilateral.

Other facts

• 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 ([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.

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.

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.

See also

• Pythagorean theorem
• Square lattice
• Square tiling
• Unit square

External links

• Square from MathWorld
• Definiton and properties of a square With interactive applet
• Animated applet illustrating the area of a square