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In Euclidean 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. Characterizations A convex quadrilateral is a square if and only if it is any one of the following:A rectangle with two adjacent equal sidesA rhombus with a right vertex angleA rhombus with all angles equalA parallelogram with one right vertex angle and two adjacent equal sidesA quadrilateral with four equal sides and four right anglesA 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}). } :Corollary 15 Properties 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 being 360/4 = 90, 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.
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