In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted
◻
{\displaystyle \square }
ABCD.
A convex 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 fourright 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
=
1
2
(
a
2
+
c
2
)
=
1
2
(
b
2
+
d
2
)
.
{\displaystyle A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2}).}
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:
The diagonals of a square bisect each other and meet at 90°
The diagonals of a square bisect its angles.
Opposite sides of a square are both parallel and equal in length.
All four angles of a square are equal. (Each is 360°/4 = 90°, so every angle of a square is a right angle.)
All four sides of a square are equal.
The diagonals of a square are equal.
The square is the n=2 case of the families of n-hypercubes and n-orthoplexes.