Square (geometry)

In plane geometry, a square is a polygon Polygon

A polygon is a closed [i] planar [i] path composed of a finite number of sequential ...

with four equal sides, four right angles, and parallel opposite sides.

Discussions

Discussion Features

Ask a question about 'Square (geometry)'

Start a new discussion about 'Square (geometry)'

Answer questions about 'Square (geometry)'

'Square (geometry)' discussion forum

Encyclopedia

In plane geometry, a square is a polygon Polygon

A polygon is a closed [i] planar [i] path composed of a finite number of sequential ...

with four equal sides, four right angles, and parallel opposite sides.

Classification

A square is a special case of a regular quadrilateral Quadrilateral

In geometry [i], a quadrilateral is a polygon [i] with four sides and four vertices. ...

, rectangle Rectangle

In geometry [i], a rectangle is defined as a quadrilateral [i] where all four of its angles are right angle [i] ...

, rhombus Rhombus

In geometry [i], a rhombus is a quadrilateral [i] in which all of the sides are of equal length, i.e., i ...

, kite Kite

A kite is a flying tethered man-made object....

, parallelogram Parallelogram

A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides....

, and isosceles trapezoid/trapezium Trapezoid

A trapezoid or trapezium is a quadrilateral [i] two of whose sides are parallel [i] to ea ...

.

Mensuration formulae

The perimeter of a square whose sides have length s is
And the area Area

Area is a physical quantity [i] expressing the size of a part of a surface [i]. ...

is

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 , while the interior of the same consists of all points 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 Rhombus

In geometry [i], a rhombus is a quadrilateral [i] in which all of the sides are of equal length, i.e., i ...

are equal, then that rhombus must be a square. The diagonals of a square are times the length of a side of the square. This value, known as Pythagoras’ constant Square root of 2

The square root [i] of 2, also known as Pythagoras' constant, often denoted by
...

, was the first number known to be irrational.

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

Other facts

If a circle is circumscribed around a square, the area of the circle is times the area of the square.
If a circle is inscribed in the square, the area of the circle is times the area of the square.
A square has a larger area than any other quadrilateral with the same perimeter .
A square is one of three regular polygon Regular polygon

A regular polygon is a simple polygon [i] which is [i] and equilateral [i] ...

s that can form a regular tiling Tessellation

A tessellation or tiling of the plane [i] is a collection of plane figure [i]s that fills th ...

of the plane . This is a consequence of the fact that the measure of the angles is a divisor of 360°.
The square is both the measure polytope Measure polytope

In geometry [i], a measure polytope is an n-dimensional analogue of a square [i] and a cube [i] ...

and the cross polytope Cross-polytope

In geometry [i], a cross-polytope, or orthoplex, is a regular [i], convex polytope [i] ...

in two dimensions. The Schläfli symbol Schläfli symbol

In mathematics [i], the Schlfli symbol is a simple notation that gives a summary of some important prope ...

for the square is .
The square is a highly symmetric object. There are four lines of reflectional symmetry Reflection symmetry

Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry [i] ...

and it has rotational symmetry Rotational symmetry

Rotational symmetry is symmetry [i] with respect to some or all rotation [i]s in m-dimensional Euclidean space [i] ...

through 90°, 180° and 270°. Its symmetry group Symmetry group

The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ...

is the dihedral group Dihedral group

In mathematics [i], the dihedral group of order [i] 2n is the abstract group of which one repr ...

D4.
If the area of a given square with side length S is multiplied by the area of a "unit triangle" , which is /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 Great circle

A great circle is a circle [i] on the surface of a sphere [i] that has the same circumference as the sph ...

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 Hyperbolic geometry

Hyperbolic geometry is a non-Euclidean geometry [i], meaning that the parallel postulate [i] of Euclidean geometry [i] ...

, 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 Finite geometry

A finite geometry is any geometric [i] system that has only a finite [i] number of points [i] ...

, a subdivided p×p square, with p a prime number, provides a model for a finite geometry with p² points. See .

External links

from MathWorld
With interactive applet