# Square Research Materials

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Rihanna - Rehab ft. Justin Timberlake
Music video by Rihanna performing Rehab. YouTube view counts pre-VEVO: 19591123. (C) 2007 The Island Def Jam Music Group.
Red vs. Blue S8 Tex fights Reds and Blues in awesome action sequence
Go to RoosterTeeth.com for all of season 8 of RvB!
Einstein vs Stephen Hawking -Epic Rap Battles of History #7
Download this song: http://bit.ly/EpicRap7 New ERB merch: http://bit.ly/MNwYxq Tweet this Vid-ee-oh: http://clicktotweet.com/TpUg9 Hi. My name is Nice Peter,...
Rihanna - We Ride
Music video by Rihanna performing We Ride. (C) 2006 The Island Def Jam Music Group.
One Direction - One Way Or Another (Teenage Kicks)
As well as releasing the Red Nose Day single, One Direction are fundraising by doing something funny for money...and they want you to join them! Get involved...
MACKLEMORE & RYAN LEWIS - CAN'T HOLD US FEAT. RAY DALTON (OFFICIAL MUSIC VIDEO)
Macklemore & Ryan Lewis present the official music video for Can't Hold Us feat. Ray Dalton. Can't Hold Us on iTunes: https://itunes.apple.com/us/album/cant-...
Draw My Life- Jenna Marbles
This video accidentally turned out kind of sad, ME SO SOWWY IT NOT POSED TO BE SAD WHO WANTS HUGS AND COOKIES? Also, FYI for anyone attempting this, it takes...
Rihanna - Pon de Replay (Internet Version)
Music video by Rihanna performing Pon de Replay. YouTube view counts pre-VEVO: 4166822. (C) 2005 The Island Def Jam Music Group.
Key & Peele: Substitute Teacher
A substitute teacher from the inner city refuses to be messed with while taking attendance.
Draw My Life - Ryan Higa
So i was pretty hesitant to make this video... but after all of your request, here is my Draw My Life video! Check out my 2nd Channel for more vlogs: http://...
David Guetta - Just One Last Time ft. Taped Rai
"Just One Last Time" feat. Taped Rai. Available to download on iTunes including remixes of : Tiësto, HARD ROCK SOFA & Deniz Koyu http://smarturl.it/DGJustOne...
Harrison Ford Won't Answer Star Wars Questions
See Harrison Ford in 42! Go to http://42movie.warnerbros.com/ Jimmy Kimmel Live - Harrison Ford Won't Answer Star Wars Questions Jimmy Kimmel Live's YouTube ...
Square

Type Regular polygon
Edges and vertices 4
Schläfli symbol {4}
Coxeter diagram
Symmetry group D4, order 2×4
Internal angle (degrees) 90°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles (90-degree 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 $\square$ ABCD.

The square is the n=2 case of the families of n-hypercubes and n-orthoplexes.

## 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 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 quadrilateral where the diagonals are equal and are the perpendicular bisectors of each other, i.e. a rhombus with equal diagonals

## Perimeter and area

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

The perimeter of a square whose four sides have length $\ell$ is

$P=4\ell$

and the area A is

$A=\ell^2.$

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

$A=\frac{d^2}{2}.$

In terms of the circumradius R, the area of a square is

$A=2R^2$

and in terms of the inradius r, its area is

$A=4r^2.$

## Coordinates and equations

$|x| + |y| = 2$ plotted on Cartesian 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 (xi, yi) with −1 < xi < 1 and −1 < yi < 1.

The equation

$\max(x^2, y^2) = 1$

describes a square of side 2, centered at the origin. 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 $\scriptstyle \sqrt{2}$. 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 a square with the center coordinates of (a, b) and a horizontal or vertical radius of r.

## Construction

Construction of a square using a compass and straightedge.

The animation at the right shows how to construct a square using a compass and straightedge.

## Properties

A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a parallelogram (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:[4]

• The diagonals of a square bisect each other and meet at 90°
• The diagonals of a square bisect its angles.
• The diagonals of a square are perpendicular.
• 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.

## Other facts

• The diagonals of a square are $\scriptstyle \sqrt{2}$ (about 1.414) times the length of a side of the square. This value, known as 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 $\pi/2$ (about 1.571) times the area of the square.
• If a circle is inscribed in the square, the area of the circle is $\pi/4$ (about 0.7854) times the area of the square.
• A square has a larger area than any other quadrilateral with the same perimeter.[5]
• 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 {4}.
• 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 D4.
• 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,[6]
$2(PH^2-PE^2) = PD^2-PB^2.$

## 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. 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.

Examples:

 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 Euclidean plane with 4 around each vertex, with each square having an internal angle of 90°. The Schläfli symbol is {4,4}. 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.

## Graphs

3-simplex (3D)

The K4 complete graph is often drawn as a square with all 6 edges connected. This graph also represents an orthographic projection of the 4 vertices and 6 edges of the regular 3-simplex (tetrahedron).

## References

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