Square (geometry)

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

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 $\sqrt{2}$ (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 $\pi/\sqrt{2}$ (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 {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.

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

Geometric shapes | Quadrilaterals

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