Square (geometry)

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Image:SQUARE SHAPE.svg Image:SquareProperties.JPG

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

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

Classification

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

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Mensuration formulae

The perimeter of a square whose sides have length s is

And the area is

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.

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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.

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Properties

The angles of a square are necessarily right angles.

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 <math>\sqrt{2}</math> (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.

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Other facts

If a circle is circumscribed around a square, the area of the circle is <math>\pi/\sqrt{2}</math> (about 2.22) times the area of the square.
If a circle is inscribed in the square, the area of the circle is <math>\pi/4</math> (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 is a factor of 360°.
The square is both the measure polytope and the cross polytope in two dimensions. The Schläfli symbol for the square is {4}.

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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.

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