Building Squares

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 ◻ ABCD.

Price

The square is priced at $4.

Characterizations

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 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 = 1/2(a² + c²) = 1/2(b² + d²)

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 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.
• A square has Schläfli symbol. A truncated square, t, is an octagon. An alternated square, h, is a digon.

Other facts

• A square can also be defined as a parallelogram with equal diagonals that bisect the angles.
• If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square.
• If a circle is circumscribed around a square, the area of the circle is π / 2 (about 1.5708) times the area of the square.
• If a circle is inscribed in the square, the area of the circle is π / 4 (about 0.7854) 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 square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry of order 4 (through 90°, 180° and 270°). Its symmetry group is the dihedral group D4.

Content from wikipedia