Square (geometry)

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In plane geometry, a square is a polygon with four equal sides and four right angles.

Contents

1 Classification
2 Mensuration formulae
3 Standard coordinates
4 Properties
5 Other facts
6 Non-Euclidean geometry
7 Finite geometry
8 See also
9 External links

Classification

Squares are a special case of regular quadrilateral, rectangle, rhombus, kite, parallelogram, and isosceles trapezoid/trapezium.

Mensuration formulae

The perimeter of a square whose sides have length s is

[P=4s]

And the area is

[A=s^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.

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 < x_i < 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] (about 1.41) times the length of a side of the square. This value, known as Pythagoras’ constant was the first number known to be irrational.

If a figure is both a rectangle and a rhombus, then it is a square.

Other facts

If a circle is circumscribed around a square, the area of the circle is [\pi/\sqrt] (about 2.22) times the area of the square.
If a circle is inscribed in the square, the area of the circle is [\pi/4] (about 0.79) times the area of the square.
A square has a larger area than any other rectangle or rhombus with the same perimeter.
A square is one of three regular polygons that can form a regular tiling of the plane (the others are the equilateral triangle and the regular hexagon). This is a consequence of the fact that the measure of the angles (90°) is a divisor of 360°.
The square is both the measure polytope and the cross polytope in two dimensions. The Schläfli symbol for the square is .
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.

Non-Euclidean geometry

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.

Finite geometry

In finite geometry, a subdivided q×q square, with q a power of a prime number, provides a model for a finite geometry with q² points. See [finite geometry of the square and cube].

External links

[Triangle with two squares] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
[Square] from MathWorld

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