, a square is a regular quadrilateral
, which means that it has four equal sides and four equal angle
angles, or 100-gradian
angles or right angles
). It can also be defined as a rectangle
in which two adjacent sides have equal length. A square with vertices
''ABCD'' would be denoted .
A convex quadrilateral
is a square if and only if
it is any one of the following:
* A rectangle
with two adjacent equal sides
* A rhombus
with a right vertex angle
* A rhombus with all angles equal
* A parallelogram
with one right vertex angle and two adjacent equal sides
* A quadrilateral
with four equal sides and four right angle
* A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals)
* A convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is
"Properties of equidiagonal quadrilaterals"
''Forum Geometricorum'', 14 (2014), 129-144.
A square is a special case of a rhombus
(equal sides, opposite equal angles), a kite
(two pairs of adjacent equal sides), a trapezoid
(one pair of opposite sides parallel), a parallelogram
(all opposite sides parallel), a quadrilateral
or tetragon (four-sided polygon), and a rectangle
(opposite sides equal, right-angles), and therefore has all the properties of all these shapes, namely:
* The diagonal
s of a square bisect
each other and meet at 90°.
* The diagonals of a square bisect its angles.
* Opposite sides of a square are both parallel
and equal in length.
* All four angles of a square are equal (each being 360°/4 = 90°, a right angle).
* All four sides of a square are equal.
* The diagonals of a square are equal.
* The square is the n=2 case of the families of n-hypercubes
* A square has Schläfli symbol
. A truncated
square, t, is an octagon
, . An alternated
square, h, is a digon
Perimeter and area
150px|The area of a square is the product of the length of its sides.
of a square whose four sides have length
and the area
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 area can also be calculated using the diagonal ''d'' according to
In terms of the circumradius
''R'', the area of a square is
since the area of the circle is
the square fills approximately 0.6366 of its circumscribed circle
In terms of the inradius
''r'', the area of the square is
Because it is a regular polygon
, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if ''A'' and ''P'' are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality
with equality if and only if the quadrilateral is a square.
* The diagonals of a square are
(about 1.414) times the length of a side of the square. This value, known as the square root of 2
or Pythagoras' constant,
was the first number proven to be irrational
* A square can also be defined as a parallelogram
with equal diagonals that bisect the angles.
* If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square.
* If a circle is circumscribed around a square, the area of the circle is
(about 1.5708) times the area of the square.
* If a circle is inscribed in the square, the area of the circle is
(about 0.7854) 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 .
* The square is a highly symmetric object. There are four lines of reflectional symmetry
and it has rotational symmetry
of order 4 (through 90°, 180° and 270°). Its symmetry group
is the dihedral group
* If the inscribed circle of a square ''ABCD'' has tangency points ''E'' on ''AB'', ''F'' on ''BC'', ''G'' on ''CD'', and ''H'' on ''DA'', then for any point ''P'' on the inscribed circle,
is the distance from an arbitrary point in the plane to the ''i''-th vertex of a square and
is the circumradius
of the square, then
are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then
is the circumradius of the square.
Coordinates and equations
The coordinates for the vertices
of a square with vertical and horizontal sides, centered at the origin and with side length 2 are (±1, ±1), while the interior of this square consists of all points (''x''i
) with and . The equation
specifies the boundary of this square. This equation means "''x''2
, whichever is larger, equals 1." The circumradius
of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and is equal to
Then the circumcircle
has the equation
Alternatively the equation
can also be used to describe the boundary of a square with center coordinates
(''a'', ''b''), and a horizontal or vertical radius of ''r''.
The following animations show how to construct a square using a compass and straightedge
. This is possible as 4 = 22
, a power of two
The ''square'' has Dih4
8. There are 2 dihedral subgroups: Dih2
, and 3 cyclic
, and Z1
A square is a special case of many lower symmetry quadrilaterals:
* A rectangle with two adjacent equal sides
* A quadrilateral with four equal sides and four right angle
* A parallelogram with one right angle and two adjacent equal sides
* A rhombus with a right angle
* A rhombus with all angles equal
* A rhombus with equal diagonals
These 6 symmetries express 8 distinct symmetries on a square. John Conway
labels these by a letter and group order.
Each subgroup symmetry allows one or more degrees of freedom for irregular quadrilateral
s. r8 is full symmetry of the square, and a1 is no symmetry. d4 is the symmetry of a rectangle
, and p4 is the symmetry of a rhombus
. These two forms are duals
of each other, and have half the symmetry order of the square. d2 is the symmetry of an isosceles trapezoid
, and p2 is the symmetry of a kite
. g2 defines the geometry of a parallelogram
Only the g4 subgroup has no degrees of freedom, but can seen as a square with directed edge
Squares inscribed in triangles
Every acute triangle
has three inscribed
squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle
two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two ''distinct'' inscribed squares. An obtuse triangle
has only one inscribed square, with a side coinciding with part of the triangle's longest side.
The fraction of the triangle's area that is filled by the square is no more than 1/2.
Squaring the circle
Squaring the circle
, proposed by ancient geometers
, is the problem of constructing a square with the same area as a given circle
, by using only a finite number of steps with compass and straightedge
In 1882, the task was proven to be impossible as a consequence of the Lindemann–Weierstrass theorem
, which proves that pi
() is a transcendental number
rather than an algebraic irrational number
; that is, it is not the root
of any polynomial
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. Larger spherical squares have larger angles.
In hyperbolic geometry
, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger hyperbolic squares have smaller angles.
A crossed square is a faceting
of the square, a self-intersecting polygon created by removing two opposite edges of a square and reconnecting by its two diagonals. It has half the symmetry of the square, Dih2
, order 4. It has the same vertex arrangement
as the square, and is vertex-transitive
. It appears as two 45-45-90 triangle
with a common vertex, but the geometric intersection is not considered a vertex.
A crossed square is sometimes likened to a bow tie
. the crossed rectangle
is related, as a faceting of the rectangle, both special cases of crossed quadrilateral
The interior of a crossed square can have a polygon density
of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
A square and a crossed square have the following properties in common:
* Opposite sides are equal in length.
* The two diagonals are equal in length.
* It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
It exists in the vertex figure
of a uniform star polyhedra
, the tetrahemihexahedron
The K4 complete graph
is often drawn as a square with all 6 possible edges connected, hence appearing as a square with both diagonals drawn. This graph also represents an orthographic projection
of the 4 vertices and 6 edges of the regular 3-simplex
* Pythagorean theorem
* Square lattice
* Square number
* Square root
* Squaring the square
* Unit square
Animated course (Construction, Circumference, Area)
With interactive applet
Category:Types of quadrilaterals