Square

This article is about the shape. For other uses, see Square (disambiguation).

Square
A regular quadrilateral
Type	Regular polygon
Edges and vertices	4
Schläfli symbol	{4}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D4), order 2×4
Internal angle (degrees)	90°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, with four equal angles, and of rhombuses, with four equal sides. As with rectangles, a square's angles are right angles (90 degrees, or π/2 radians).

Characterizations

A quadrilateral is a square if and only if it is any one of the following:

• A rectangle with four 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 angles; that is, a quadrilateral that is both a rhombus and a rectangle
• 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 $A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2})$
• A parallelogram with equal diagonals that bisect the angles

Squares are the only regular polygons whose internal angle, central angle, and external angle are all equal (they are all right angles).

Properties

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:

• All four internal angles of a square are equal (each being 360°/4 = 90°, a right angle).
• The central angle of a square is equal to 90° (360°/4).
• The external angle of a square is equal to 90°.
• The diagonals of a square are equal and bisect each other, meeting at 90°.
• The diagonal of a square bisects its internal angle, forming adjacent angles of 45°.
• All four sides of a square are equal.
• Opposite sides of a square are parallel.

Measurement

The area of a square is the product of the lengths of its sides.

A square whose four sides have length $\ell$ has perimeter $P=4\ell$ and diagonal length $d={\sqrt {2}}\ell$. (The square root of 2 is irrational, meaning that it is not the ratio of any two integers. It is approximately equal to 1.414.) Its area is $$A=\ell ^{2}={\tfrac {1}{2}}d^{2}.$$

Since four squared equals sixteen, a four by four square has an area equal to its perimeter. That is, it is an equable shape. The only other integer rectangle with such a property is a three by six rectangle.

In classical times, the second power was described in terms of the area of a square, as in the formula for area from side length. This led to the use of the term square to mean raising to the second power.

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 holds: $$16A\leq P^{2}$$ with equality if and only if the quadrilateral is a square.

Related shapes

See also: Cyclic quadrilateral and Tangential quadrilateral

The inscribed circle of a square is the largest circle that can fit inside that square. Its center is at the center point of the square, and its radius (the inradius of the square) has length $r=\ell /2$. The inscribed circle touches the sides of the square at their midpoints. The circumscribed circle of a square is the circle passing through the four vertices. Its radius, known as the circumradius, has length $R=\ell /{\sqrt {2}}$. 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,$$2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.$$ If $d_{i}$ is the distance from an arbitrary point in the plane to the i-th vertex of a square and $R$ is the circumradius of the square, then$${\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.$$ If $L$ and $d_{i}$ are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then$$d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})$$ and $$d_{1}^{2}d_{3}^{2}+d_{2}^{2}d_{4}^{2}=2(R^{4}+L^{4}),$$ where $R$ is the circumradius of the square.

A square tiling is one of three regular tilings of the plane. The others are the tilings made from the equilateral triangle and the regular hexagon.

Because a square is a convex regular polygon with four sides, its Schläfli symbol is {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}. The square is part of two infinite families of regular polytopes, one which includes the cube in three dimensions and the hypercubes in higher dimensions, and another which includes the regular octahedron in three dimensions and the cross polytopes in higher dimensions.

Coordinates and equations

$|x|+|y|=2$ plotted on Cartesian coordinates.

A unit square is a square of side length one. Often it is represented in Cartesian coordinates as the square enclosing the points $(x,y)$ that have $0\leq x\leq 1$ and $0\leq y\leq 1$.

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, y_i) with −1 < x_i < 1 and −1 < y_i < 1. The equation $$\max(x^{2},y^{2})=1$$ specifies the boundary of this square. This equation means "x² or y², 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 ${\sqrt {2}}.$ Then the circumcircle has the equation $$x^{2}+y^{2}=2.$$

Alternatively the equation $$\left|x-a\right|+\left|y-b\right|=r.$$ 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 square is therefore the shape of a topological ball according to the L₁ distance metric.

Construction

The construction of a square with a given side, using a compass and straightedge, is given in Euclid's Elements. The existence of this construction means that squares are constructible polygons. A regular $n$-gon is constructible exactly when the odd prime factors of $n$ are distinct Fermat primes, and in the case of a square $n=4$ has no odd prime factors so this condition is vacuously true.

Square at a given circumcircle

Square at a given side length,
right angle by using Thales' theorem

Square at a given diagonal

Symmetry

Classification of quadrilaterals by their symmetry subgroups. The 8-fold symmetry of the square is labeled as r8, at the top of the image. The "gyrational square" below it corresponds to the subgroup of four orientation-preserving symmetries of a square, using rotations but not reflections.

The square is the most symmetrical of the quadrilaterals. There are eight congruence transformations of the plane that take the square to itself:

• Leaving the square unchanged (the identity transformation)
• Rotation around the center of the square by 90°, 180°, or 270°
• Reflection across a diagonal, or across a centerline of the square, parallel to one of its sides.

Combining any two of these transformations by performing one after the other produces another symmetry. This operation on pairs of symmetries gives the eight symmetries of a square the mathematical structure of a point group, the dihedral group of order eight.

Inscribed squares

Main article: Inscribed square problem

A square is said to be inscribed in a curve whenever all four vertices of the square lie on the curve. It is an unsolved problem in mathematics, the inscribed square problem, whether every simple closed curve has an inscribed square. Some cases of this problem are known to be true; for instance, it is true for every smooth curve. For instance, a square can be inscribed on any circle, which becomes its circumscribed circle. As another special case of the inscribed square problem, a square can be inscribed on the boundary of any convex set. The only other regular polygon with this property is the equilateral triangle. More strongly, there exists a convex set on which no other regular polygon can be inscribed.

The Calabi triangle and the three placements of its largest square. The placement on the long side of the triangle is inscribed; the other two are not.

For an inscribed square in a triangle, at least one of the square's sides lies on a side of the triangle. Every acute triangle has three inscribed squares, one lying on each of its three sides. In a right triangle there are two inscribed squares, one touching the right angle of the triangle and the other lying on the opposite side. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side.

An inscribed square can cover at most half the area of the triangle it is inscribed into. It is exactly half only when the square lies on a side of the triangle whose length equals the height of the triangle over that side. For a square that lies on a triangle side of length $x$, with height $h$, the square's side length will be $xh/(x+h)$. For any two inscribed squares in a triangle, the square that lies on the longer side of the triangle will have smaller area. In an acute triangle, the three inscribed squares have side lengths that are all within a factor of ${\tfrac {2}{3}}{\sqrt {2}}\approx 0.94$ of each other. The Calabi triangle, an obtuse triangle, shares with the equilateral triangle the property of having three different ways of placing the largest square that fits into it, but (because it is obtuse) only one of these three is inscribed.

Squaring the circle

A circle and square with the same area

Main article: 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 (π) is a transcendental number rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.

Packing

Main articles: Square packing, Circle packing in a square, and Squaring the square

The smallest known square that can contain 11 unit squares has side length approximately 3.877084.

Several packing problems concern squares. For instance, the problem of square packing in a square or circle asks how many unit squares can fit without overlap into a larger square or circle of a given size. Alternatively and equivalently, one may ask to minimize the size of a larger square or circle that contains a given number of unit squares, or to minimize the area left uncovered by the unit squares. For packing unit squares into a larger square of integer side length, a number of unit squares equal to the squared side length can be packed, leaving no uncovered area, but beyond a few special cases such as this one the optimal solutions to these problems remain unsolved. Circle packing in a square is another packing problem with a similar flavor, where the goal is to fit as many unit circles as possible into a larger square of a given size, or again to minimize the uncovered area. Testing whether a given number of axis-aligned unit squares can be packed into an orthogonally convex rectilinear polygon with half-integer vertex coordinates is NP-complete.

As well as problems of packing equal-size squares, mathematicians have studied several problems of packing unequal squares. The problem of squaring the square involves subdividing a given square into squares, with no uncovered area. If the subdivided squares all have distinct sizes, the result is called a perfect squared square. Solutions with as few as 21 distinct sizes are known. A relaxed version of the problem, called "Mrs. Perkins's quilt", asks for a subdivision of a square with integer side lengths into as few as possible smaller squares for which the greatest common divisor of the side lengths is 1. The optimal number of smaller squares is proportional to the logarithm of the side length of the outer square. The entire Euclidean plane can be packed by squares of all integer side lengths, one square per side length, with no uncovered area.

Non-Euclidean geometry

Non-Euclidean geometry also has polygons with four equal sides and equal angles. These are often called squares, but some authors avoid calling them that, instead calling them regular quadrilaterals, because unlike Euclidean squares they cannot have right 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 Euclidean geometry, spherical squares have obtuse angles, 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 acute angles, less than right angles. Larger hyperbolic squares have smaller angles.

Examples:

Two squares can tile the sphere with 2 squares around each vertex and 180-degree internal angles. Each square covers an entire hemisphere and their vertices lie along a great circle. This is an example of a dihedron. The Schläfli symbol is {4,2}.

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 hyperbolic plane with 5 around each vertex, with each square having 72-degree internal angles. The Schläfli symbol is {4,5}. In fact, for any n ≥ 5 there is a hyperbolic tiling with n squares about each vertex.

See also

• Pythagorean theorem
• Square lattice
• Square number
• Square root
• Squircle

External links

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