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.
12(a2+c2)(b2+d2).
A rectangle is a special case of both parallelogram and trapezoid. A square is a special case of a rectangle.
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.
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).
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°).
The dual polygon of a rectangle is a rhombus, as shown in the table below.
| Rectangle | Rhombus |
|---|---|
| All angles are equal. | All sides are equal. |
| Alternate sides are equal. | Alternate angles are equal. |
| Its centre is equidistant from its vertices, hence it has a circumcircle. | Its centre is equidistant from its sides, hence it has an incircle. |
| Two axes of symmetry bisect opposite sides. | Two axes of symmetry bisect opposite angles. |
| Diagonals are equal in length. | Diagonals intersect at equal angles. |
| All angles are right angles; opposite sides are equal and parallel | All sides are equal; opposite sides are parallel. |
- The figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa.
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.
The formula for the perimeter of a rectangle
The area of a rectangle is the product of the length and width.
If a rectangle has length A=ℓw;
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.
The Japanese theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle.
The British flag theorem states that with vertices denoted A, B, C, and D, for any point P on the same plane of a rectangle:
a and b, with two ways of being folded along a line through its center such that the area of overlap is minimized and each area yields a different shape
A perfect rectangle of order 9
Lowest-order perfect squared square (1) and the three smallest perfect squared squares (2–4) – Coxeter, Harold Scott MacDonald; Longuet-Higgins, M.S.; Miller, J.C.P. (1954). "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. 246 (916). The Royal Society: 401–450. Bibcode:1954RSPTA.246..401C. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446. S2CID 202575183.