In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as .
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle).
A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.
Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.
Characterizations
A convex quadrilateral is a rectangle if and only if it is any one of the following: a parallelogram with at least one right angle a parallelogram with diagonals of equal length a parallelogram ABCD where triangles ABD and DCA are congruent an equiangular quadrilateral a quadrilateral with four right angles a quadrilateral where the two diagonals are equal in length and bisect each other a convex quadrilateral with successive sides a, b, c, d whose area is . a convex quadrilateral with successive sides a, b, c, d whose area is
Classification
Traditional hierarchy
A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular.
A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which both pairs of opposite sides are parallel and equal in length.
A trapezium is a convex quadrilateral which has at least one pair of parallel opposite sides.
A convex quadrilateral is
Simple: The boundary does not cross itself. Star-shaped: The whole interior is visible from a single point, without crossing any edge.
Alternative hierarchy
De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects.
Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia (crossed quadrilaterals with the same vertex arrangement as isosceles trapezia).
Properties
Symmetry
A rectangle is cyclic: all corners lie on a single circle.
It is equiangular: all its corner angles are equal (each of 90 degrees).
It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit.
It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
Rectangle-rhombus duality
The dual polygon of a rectangle is a rhombus, as shown in the table below.
The figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa.
Miscellaneous
A rectangle is a rectilinear polygon: its sides meet at right angles.
A rectangle in the plane can be defined by five independent degrees of freedom consisting, for example, of three for position (comprising two of translation and one of rotation), one for shape (aspect ratio), and one for overall size (area).
Two rectangles, neither of which will fit inside the other, are said to be incomparable.
Formulae
If a rectangle has length and width
it has area , it has perimeter , each diagonal has length , and when , the rectangle is a square.
Theorems
The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area.
The midpoints of the sides of any quadrilateral with perpendicular diagonals form a rectangle.
A parallelogram with equal diagonals is a rectangle.