Square

Hierarchy of the squares

Square is one Figure the even Geometry, indeed a polygon with four corners and four sides. In mathematics one (even) defines squares as Polygone with four Corners, and (therefore it also) four Edges (or sides).

That regular (or regular) Square is that Square.

A square has two Diagonal one. Couches both diagonals within the square, like that is the square convex (convex square), lies exactly one diagonal outside, so the square has one concave Corner (non-convex square). With one estimated (also: ) both diagonals are appropriate for square outside of the square. Estimated squares are generalized Polygone and normally to (normal or "genuine") the squares are not counted. Same applies to degenerated Squares, with those two or more corner points collapse or more than 2 corner points on a straight line lie.

To each square applies:

• The winkelsumme (more exact: those Internal angle sum) in (not estimated) a square 360 degrees amount to and/or. 2?.

(with estimated squares there is none "inside" and none "outside", and therefore also no internal angles!)

• It is sample tile for a periodic Parqueting (Euclidean) the level (Space filler).

Table of contents 1 Special squares 2 Classification 3 Formulas 4 See also

Special squares

• Trapezoid: Square with (at least) two parallel sides.

• Parallelogram: Square, with ever two each other opposite sides parallel are.

• Rectangle: Square with four equal large (inside)Winkeln (90°, see right angle)

• Deltoid (Kite square): Square, with that the diagonals stand perpendicularly one on the other and a diagonal by the other one are halved. <== > Square with two pairs of neighbouring sides, equivalent are in each case long.

• Rhombus (Lozenge): Square with four equal long sides
• Square: Rectangle with four equal long pages <== > Equal to rhombus with four to angles
• Chord square: Square with one Periphery (the four sides are chords of the Umkreises.)
• Tangent square: Square with one Inkreis (the four sides are tangents to the inkreis.)

Between the individual square types u apply.A. the following quantity ratios:
(each term stands X synonymously for All X mixes)

The subset relations represented in the diagram, for example:
Square? Rectangle? Parallelogram? Trapezoid? Konvexes_Viereck

Square = rectangle? Lozenge
Square = kite square? equal-leg trapezoid
Rectangle = chord square? Parallelogram
Lozenge = kite square? Trapezoid
Lozenge = tangent square? Parallelogram
Gleichschenkliges_Trapez = chord square? Trapezoid

Classification

The even squares are divided on the basis of different criteria:

• after characteristics of the inside:
  • convex
  • not convex

• after Symmetry characteristics:
  • a diagonal is symmetry axis: Deltoid
  • both diagonals are symmetry axes: Rhombus
  • a Seitensymmetrale: gleichschenkeliges trapezoid
  • two Seitensymmetralen: Rectangle
  • four symmetry axes: Square
  • zweizaehlige symmetry (point symmetrical): Parallelogram
  • vierzaehlige symmetry: Square

• after the length of the sides:
  • two pairs of equivalent long opposite sides: Parallelogram
  • two pairs of equivalent long neighbouring sides: Deltoid
  • equilateral square: Rhombus
  • the sum of the lengths of opposite sides is alike: Tangent square

• after the size of the angles:
  • two pairs equivalent large opposite angle: Parallelogram
  • two pairs equivalent large neighbouring angle: gleichschenkeliges trapezoid
  • gleichwinkeliges square: Rectangle
  • the sum of opposite angles results in 180°: Chord square

• according to the situation of the sides:
  • a pair of parallel sides: Trapezoid
  • two pair of parallel sides: Parallelogram
  • the sides affect the same circle (the inkreis): Tangent square

• according to the situation of corners:
  • the corners are on a circle (the periphery): Chord square

Formulas

Designations at the square

Internal angle sum is 360°: <math> \alpha+ \beta+ \gamma+ \delta=360^ \circ</math>

<math> \theta = 90^ \circ \Longleftrightarrow a^2+c^2 = b^2+d^2</math>

<math>A = \frac{1}{2} e f \sin \theta</math>

<math>A = \frac{1}{4} \left(b^2+d^2-a^2-c^2 \right) \tan \theta</math>

<math>A = \frac{1}{4} \sqrt{4e^2f^2 - \left(b^2+d^2-a^2-c^2 \right)^2}</math>

A square can be described among other things by suitable combinations of the following instructions (five from each other independent data):

• Angle at the corners (internal angles)
• Length of the sides
• Length of the diagonals
• Extent
• Surface

See also

- word origin, Synonymous one and translations

• Inequations in squares
• Mathematics for the school