Square (geometry)

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Square
Square diagonals.svg
A square
The sides of a square and its diagonals meet at right angles.
Edges and vertices 4
Schläfli symbols {4}
{}x{}
Coxeter–Dynkin diagrams CDW ring.pngCDW 4.pngCDW dot.png
Image:CDW_ring.pngCDW 2.pngImage:CDW_ring.png
Symmetry group Dihedral (D4)
Area
(with t=edge length)
t2
Internal angle
(degrees)
90°

In plane (Euclidean) geometry, a square is a regular polygon with four sides. It may also be thought of as a special case of a rectangle, as it has four right angles and parallel sides. Likewise, it is also a special case of a rhombus, kite, parallelogram, and trapezoid.

Contents

Mensuration formulae

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

The perimeter of a square whose sides have length t is

P = 4t.

And the area is

A = t2.

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.

Standard 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 (x0, x1) with −1 < xi < 1.

Properties

Each angle in a square is equal to 90 degrees, or a right angle.

The diagonals of a square are equal. Conversely, if the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are \sqrt{2} (about 1.41) times the length of a side of the square. This value, known as Pythagoras’ constant, was the first number proven to be irrational.

If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths) then it is a square.

Other facts

  • If a circle is circumscribed around a square, the area of the circle is π / 2 (about 1.57) times the area of the square.
  • If a circle is inscribed in the square, the area of the circle is π / 4 (about 0.79) 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 {4}.
  • The square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry through 90°, 180° and 270°. Its symmetry group is the dihedral group D4.
  • If the area of a given square with side length S is multiplied by the area of a "unit triangle" (an equilateral triangle with side length of 1 unit), which is \tfrac{\sqrt{3}}{4} units squared, the new area is that of the equilateral triangle with side length S.

Non-Euclidean geometry

In non-euclidean geometry, squares are more generally polygons with four 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.

In hyperbolic geometry, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger squares have smaller angles.

Examples:

Square on sphere.png
Six squares can tile the sphere with three squares around each vertex and 120 degree internal angles. This is called a spherical cube. The Schläfli symbol is {4,3}.
Square on plane.png
Squares can tile the Euclidean plane with four around each vertex, with each square having an internal angle of 90 degrees. The Schläfli symbol is {4,4}.
Square on hyperbolic plane.png
Squares can tile the hyperbolic plane with five around each vertex, with each square having 72 degree internal angles. The Schläfli symbol is {4,5}.

See also

  • Pythagorean theorem
  • Square lattice
  • Square tiling
  • Unit square

References

  • Arnone, Wendy. 2001. Geometry for Dummies. New York, NY: Hungry Minds. ISBN 0764553240
  • Hartshorne, Robin. 2002. Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. New York: Springer. ISBN 0387986502
  • Leff, Lawrence S. 1997. Geometry the Easy Way. Hauppauge, NY: Barron’s Educational Series. ISBN 0764101102
  • Stillwell, John. 2005. The Four Pillars of Geometry. Undergraduate Texts in Mathematics. New York: Springer. ISBN 0387255303

External links

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