Get Square essential facts below. View Videos or join the Square discussion. Add Square to your PopFlock.com topic list for future reference or share this resource on social media.



A convex quadrilateral is a square if and only if it is any one of the following:[3][4]

  • A rectangle with two adjacent equal sides
  • A rhombus with a right vertex angle
  • A rhombus with all angles equal
  • 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 [5]



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

Perimeter and area


The perimeter of a square whose four sides have length is

and the area A is



The area can also be calculated using the diagonal d according to




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.[7] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:

with equality if and only if the quadrilateral is a square.

Other facts

  • 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,[2] was the first number proven to be irrational.
  • A square has a larger area than any other quadrilateral with the same perimeter.[8]
  • If and are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then [11]
where is the circumradius of the square.

Coordinates and equations

plotted on Cartesian coordinates.


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 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.



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




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


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



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 with rational coefficients.

Non-Euclidean geometry





Tetragonal dihedron.png
Square on sphere.svg
Square on plane.svg
Square on hyperbolic plane.png
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






  • Opposite sides are equal in length.
  • The two diagonals are equal in length.




See also


  1. ^ "List of Geometry and Trigonometry Symbols". Math Vault. 2020-04-17. Retrieved .
  2. ^ a b c Weisstein, Eric W. "Square". mathworld.wolfram.com. Retrieved .
  4. ^ "Problem Set 1.3". jwilson.coe.uga.edu. Retrieved .
  5. ^ Josefsson, Martin, "Properties of equidiagonal quadrilaterals" Forum Geometricorum, 14 (2014), 129-144.
  6. ^ "Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram". www.mathsisfun.com. Retrieved .
  7. ^ Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
  8. ^ 1999, Martin Lundsgaard Hansen, thats IT (c). "Vagn Lundsgaard Hansen". www2.mat.dtu.dk. Retrieved .CS1 maint: numeric names: authors list (link)
  11. ^ Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335-355. arXiv:2010.12340.
  13. ^ Wells, Christopher J. "Quadrilaterals". www.technologyuk.net. Retrieved .

External links




Music Scenes