In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles).[1] It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ◻ displaystyle square ABCD. Contents 1 Characterizations 2 Properties 2.1
3 Coordinates and equations 4 Construction 5 Symmetry 6 Squares inscribed in triangles 7 Squaring the circle 8 Non-Euclidean geometry 9 Crossed square 10 Graphs 11 See also 12 References 13 External links Characterizations A convex quadrilateral is a square if and only if it is any one of the following:[2][3] 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 angles 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 = 1 2 ( a 2 + c 2 ) = 1 2 ( b 2 + d 2 ) . displaystyle A= tfrac 1 2 (a^ 2 +c^ 2 )= tfrac 1 2 (b^ 2 +d^ 2 ). [4]:Corollary 15 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:[5] The diagonals 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 is 360°/4 = 90°, so
every angle of a square is 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 and
n-orthoplexes.
A square has
The area of a square is the product of the length of its sides. The perimeter of a square whose four sides have length ℓ displaystyle ell is P = 4 ℓ displaystyle P=4ell and the area A is A = ℓ 2 . displaystyle A=ell ^ 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. The area can also be calculated using the diagonal d according to A = d 2 2 . displaystyle A= frac d^ 2 2 . In terms of the circumradius R, the area of a square is A = 2 R 2 ; displaystyle A=2R^ 2 ; since the area of the circle is π R 2 , displaystyle pi R^ 2 , the square fills approximately 0.6366 of its circumscribed circle. In terms of the inradius r, the area of the square is A = 4 r 2 . displaystyle A=4r^ 2 . 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.[6] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds: 16 A ≤ P 2 displaystyle 16Aleq P^ 2 with equality if and only if the quadrilateral is a square. Other facts The diagonals of a square are 2 displaystyle scriptstyle sqrt 2 (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 π / 2 displaystyle pi /2 (about 1.5708) times the area of the square. If a circle is inscribed in the square, the area of the circle is π / 4 displaystyle pi /4 (about 0.7854) times the area of the square.
A square has a larger area than any other quadrilateral with the same
perimeter.[7]
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
2 ( P H 2 − P E 2 ) = P D 2 − P B 2 . displaystyle 2(PH^ 2 -PE^ 2 )=PD^ 2 -PB^ 2 . If d i displaystyle d_ i is the distance from an arbitrary point in the plane to the i-th vertex of a square and R displaystyle R is the circumradius of the square, then[9] d 1 4 + d 2 4 + d 3 4 + d 4 4 4 + 3 R 4 = ( d 1 2 + d 2 2 + d 3 2 + d 4 2 4 + R 2 ) 2 . displaystyle 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 . Coordinates and equations
x
+
y
= 2 displaystyle x+y=2 plotted on Cartesian coordinates. 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 (xi, yi) with −1 < xi < 1 and −1 < yi < 1. The equation max ( x 2 , y 2 ) = 1 displaystyle 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 equals 2 displaystyle scriptstyle sqrt 2 . Then the circumcircle has the equation x 2 + y 2 = 2. displaystyle x^ 2 +y^ 2 =2. Alternatively the equation
x − a
+
y − b
= r . displaystyle leftx-aright+lefty-bright=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. Construction The following animations show how to construct a square using a compass and straightedge. This is possible as 4 = 22, a power of two.
Symmetry The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the square is r12 and no symmetry is labeled a1. The square has Dih4 symmetry, order 8. There are 2 dihedral subgroups: Dih2, Dih1, and 3 cyclic subgroups: Z4, Z2, 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 angles 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.[10]
Each subgroup symmetry allows one or more degrees of freedom for
irregular quadrilaterals. 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 edges.
Squares inscribed in triangles
Main article:
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 called a
spherical square dihedron. The
Six squares can tile the sphere with 3 squares around each vertex and
120-degree internal angles. This is called a spherical cube. The
Squares can tile the
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. Crossed square Crossed-square 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 or butterfly. the crossed rectangle is related, as a faceting of the rectangle, both special cases of crossed quadrilaterals.[11] 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. Graphs
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
Cube
Pythagorean theorem
References ^ W., Weisstein, Eric. "Square". mathworld.wolfram.com. Retrieved
2017-12-12.
^ Zalman Usiskin and Jennifer Griffin, "The Classification of
Quadrilaterals. A Study of Definition", Information Age Publishing,
2008, p. 59, ISBN 1-59311-695-0.
^ "Problem Set 1.3". jwilson.coe.uga.edu. Retrieved 2017-12-12.
^ Josefsson, Martin, "Properties of equidiagonal quadrilaterals" Forum
Geometricorum, 14 (2014), 129-144.
^ "Maths is Fun - Can't Find It (404)". www.mathsisfun.com. Retrieved
2017-12-12.
^ Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in
Mathematical Plums (R. Honsberger, editor). Washington, DC:
Mathematical Association of America, 1979: 147.
^ 1999, Martin Lundsgaard Hansen, thats IT (c). "Vagn Lundsgaard
Hansen". www2.mat.dtu.dk. Retrieved 2017-12-12.
^ "
External links Wikimedia Commons has media related to Squares (geometry). Animated course (Construction, Circumference, Area) Weisstein, Eric W. "Square". MathWorld. Definition and properties of a square With interactive applet Animated applet illustrating the area of a square v t e Polygons Regular List 1–10 sides Monogon Digon Triangle Equilateral Isosceles Quadrilateral Square Rectangle Rhombus Parallelogram Trapezoid Kite Pentagon
Hexagon
Heptagon
Octagon
11–20 sides Hendecagon Dodecagon Tridecagon Tetradecagon Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon 21–100 sides (selected)
>100 sides 120-gon
257-gon
360-gon
Star polygons (5–12 sides) Pentagram Hexagram Heptagram Octagram Enneagram Decagram Hendecagram Dodecagram v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron
Tetrahedron
Dodecahedron • Icosahedron Uniform 4-polytope
5-cell
Uniform 5-polytope
5-simplex
Uniform 6-polytope
6-simplex
Uniform 7-polytope
7-simplex
Uniform 8-polytope
8-simplex
Uniform 9-polytope
9-simplex
Uniform 10-polytope
10-simplex
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics:
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