Square (geometry)
Square | |
---|---|
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 | |
Symmetry group | Dihedral (D_{4}) |
Area (with t=edge length) |
t^{2} |
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 perimeter of a square whose sides have length t 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.
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 (x_{0}, x_{1}) 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 (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 (about 1.57) times the area of the square.
- If a circle is inscribed in the square, the area of the circle is (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 .
- 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 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:
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
All links retrieved January 1, 2020.
- Definiton and properties of a square With interactive applet.
- Animated applet illustrating the area of a square
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