Definition
In Euclidean plane geometry, a rectangle is any 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°). It can also be defined as 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 ABCD.
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (right) and angulus (angle).
A so-called crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a special case of an antiparallelogram, and its angles are not right angles. 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.
FORMULAS:
Area = w × h
w = width
h = height
Perimeter=2h x 2w
w= width
h= height
In Euclidean plane geometry, a rectangle is any 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°). It can also be defined as 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 ABCD.
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (right) and angulus (angle).
A so-called crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a special case of an antiparallelogram, and its angles are not right angles. 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.
FORMULAS:
Area = w × h
w = width
h = height
Perimeter=2h x 2w
w= width
h= height
Other rectangles
A saddle rectangle has 4 nonplanar vertices, alternated from vertices of a cuboid, with a unique minimal surface interior defined as a linear combination of the four vertices, creating a saddle surface. This example shows 4 blue edges of the rectangle, and two green diagonals, all being diagonal of the cuboid rectangular faces. In solid geometry, a figure is non-planar if it is not contained in a (flat) plane. A skew rectangle is a non-planar quadrilateral with opposite sides equal in length and four equal acute angles. A saddle rectangle is a skew rectangle with vertices that alternate an equal distance above and below a plane passing through its centre, named for its minimal surface interior seen with saddle point at its centre. The convex hull of this skew rectangle is a special tetrahedron called a rhombic disphenoid. (The term "skew rectangle" is also used in 2D graphics to refer to a distortion of a rectangle using a "skew" tool. The result can be a parallelogram or a trapezoid/trapezium.)
In spherical geometry, a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in the sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry.
In elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length.
In hyperbolic geometry, a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length.
Squared, perfect, and other tiled rectangles
A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is perfect if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is imperfect. In a perfect (or imperfect) triangled rectangle the triangles must be right triangles.
A rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares. The same is true if the tiles are unequal isosceles right triangles.
The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes.
A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is perfect if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is imperfect. In a perfect (or imperfect) triangled rectangle the triangles must be right triangles.
A rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares. The same is true if the tiles are unequal isosceles right triangles.
The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes.
Hyperrectangle
A rectangular cuboid is a 3-orthotope Type Prism Facets 2n Vertices 2n Symmetry group [2n-1], order 2n Schläfli symbol { }n Coxeter-Dynkin diagram ... Dual Rectangular n-fusil Properties convex, zonohedron, isogonal In geometry, an orthotope[1] (also called a hyperrectangle or a box) is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals.
A three-dimensional orthotope is also called a right rectangular prism, rectangular cuboid, or rectangular parallelepiped.
A special case of an n-orthotope, where all edges are equal length, is the n-hypercube.
By analogy, the term "hyperrectangle" or "box" refers to Cartesian products of orthogonal intervals of other kinds, such as ranges of keys in database theory or ranges of integers, rather than real numbers.
A rectangular cuboid is a 3-orthotope Type Prism Facets 2n Vertices 2n Symmetry group [2n-1], order 2n Schläfli symbol { }n Coxeter-Dynkin diagram ... Dual Rectangular n-fusil Properties convex, zonohedron, isogonal In geometry, an orthotope[1] (also called a hyperrectangle or a box) is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals.
A three-dimensional orthotope is also called a right rectangular prism, rectangular cuboid, or rectangular parallelepiped.
A special case of an n-orthotope, where all edges are equal length, is the n-hypercube.
By analogy, the term "hyperrectangle" or "box" refers to Cartesian products of orthogonal intervals of other kinds, such as ranges of keys in database theory or ranges of integers, rather than real numbers.
Golden Rectangle
A golden rectangle is one whose side lengths are in the golden ratio, , which is (the Greek letter phi), where is approximately 1.618.
A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same aspect ratio as the first. Square removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property.
According to astrophysicist and mathematics popularizer Mario Livio, since the publication of Luca Pacioli's Divina Proportione in 1509, when "with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use," many artists and architects have been fascinated by the presumption that the golden rectangle is considered aesthetically pleasing. The proportions of the golden rectangle have been observed in works predating Pacioli's publication.
A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship a
A golden rectangle is one whose side lengths are in the golden ratio, , which is (the Greek letter phi), where is approximately 1.618.
A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same aspect ratio as the first. Square removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property.
According to astrophysicist and mathematics popularizer Mario Livio, since the publication of Luca Pacioli's Divina Proportione in 1509, when "with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use," many artists and architects have been fascinated by the presumption that the golden rectangle is considered aesthetically pleasing. The proportions of the golden rectangle have been observed in works predating Pacioli's publication.
A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship a
Construction of Golden Rectangle
A method to construct a golden rectangle. The square is outlined in red. The resulting dimensions are in the golden ratio. A golden rectangle can be constructed with only straightedge and compass by this technique:
A method to construct a golden rectangle. The square is outlined in red. The resulting dimensions are in the golden ratio. A golden rectangle can be constructed with only straightedge and compass by this technique:
- Construct a simple square
- Draw a line from the midpoint of one side of the square to an opposite corner
- Use that line as the radius to draw an arc that defines the height of the rectangle
- Complete the golden rectangle