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Concept
Square
Summary
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted .
A quadrilateral is a square if and only if it is any one of the following:
A rectangle with two adjacent equal sides
A rhombus with a right vertex angle
A rhombus with all angles equal
A parallelogram with one right vertex angle and two adjacent equal sides
A quadrilateral with four equal sides and four right angles
A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals)
A convex quadrilateral with successive sides a, b, c, d whose area is
A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a parallelogram (all opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and a rectangle (opposite sides equal, right-angles), and therefore has all the properties of all these shapes, namely:
All four internal angles of a square are equal (each being 360°/4 = 90°, a right angle).
The central angle of a square is equal to 90° (360°/4).
The external angle of a square is equal to 90°.
The diagonals of a square are equal and bisect each other, meeting at 90°.
The diagonal of a square bisects its internal angle, forming adjacent angles of 45°.
All four sides of a square are equal.
Opposite sides of a square are parallel.
A square has Schläfli symbol {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}.
The square is the n = 2 case of the families of n-hypercubes and n-orthoplexes.
Official source
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