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Rectangle

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A rectangle is a specific type of quadrilateral, which is a polygon with four sides. In Euclidean plane geometry, it is defined as a rectilinear convex polygon with four right angles. Because all four of its angles are equal, a rectangle is also known as an equiangular quadrilateral. Each of these angles measures exactly 90 degrees.

Illustration for the area of a rectangle
Illustration for the area of a rectangle

This shape is a special case of a parallelogram. A parallelogram is a quadrilateral where opposite sides are parallel and equal in length. If a parallelogram contains at least one right angle, it must be a rectangle.

PerimeterRectangle
PerimeterRectangle

There are several ways to identify a rectangle through its geometric properties. A convex quadrilateral is a rectangle if its diagonals are equal in length and bisect each other. This means the diagonals cut each other exactly in half. Another way to define it is as a parallelogram where the diagonals are of equal length. You can also identify a rectangle if it is a parallelogram where each pair of adjacent sides is perpendicular. This perpendicular relationship is what creates the characteristic right angles.

Symmetries of square
Symmetries of square

Furthermore, the area of a rectangle is calculated as the product of its length and width. If the length and width are equal, the rectangle becomes a square.

Rectangles belong to a specific hierarchy of shapes. A rectangle is a special type of parallelogram. A parallelogram is itself a special type of trapezium, which is a convex quadrilateral with at least one pair of parallel sides. In North America, the term trapezoid is often used instead of trapezium. Within the rectangle family, a square is a special case where all four sides are of equal length. When a rectangle is not a square, it is sometimes referred to as an oblong.

Symmetries of square
Symmetries of square

The history of the word rectangle is found in the Latin language. It is derived from the combination of "rectus," meaning right or proper, and "angulus," meaning angle. This name literally describes the right angles that define the shape. Beyond standard flat geometry, different mathematical systems contain their own versions of rectangles. In spherical geometry, a spherical rectangle has edges that are great circle arcs with angles greater than 90 degrees. In hyperbolic geometry, the angles are less than 90 degrees.

Crossed rectangles
Crossed rectangles

Rectangles possess unique symmetry and duality. A rectangle is cyclic, meaning all four of its corners lie on a single circle. It also has two lines of reflectional symmetry that pass through the midpoints of opposite sides. It possesses rotational symmetry of order 2, which means it looks the same after a 180-degree turn. Interestingly, the rectangle has a duality with the rhombus. While a rectangle has equal angles, a rhombus has equal sides. If you join the midpoints of the sides of a rectangle, the resulting shape is a rhombus.

Crossed rectangles
Crossed rectangles

Mathematics also explores more complex versions, such as the crossed rectangle. A crossed rectangle is a self-intersecting quadrilateral. It is formed by two opposite sides of a rectangle and its two diagonals. This shape often resembles a bow tie or an angular eight. While it shares some properties with standard rectangles, such as equal diagonals and opposite sides, its angles are not right angles.

Crossed rectangles
Crossed rectangles

There are also saddle rectangles, which involve four nonplanar vertices that create a saddle-shaped surface in three dimensions.

Saddle rectangle example
Saddle rectangle example

Rectangles are essential in the study of tiling, or tessellation. This involves covering a plane with shapes without gaps or overlaps. Common tiling patterns using rectangles include the stacked bond, running bond, basket weave, and herringbone patterns.

Stacked bond
Stacked bond
Basketweave bond
Basketweave bond
Herringbone bond
Herringbone bond

Mathematicians also study "perfect" rectangles. A perfect rectangle is one that is tiled by smaller squares where no two squares are the same size. The lowest number of squares required to create a perfect rectangle is nine.

Perfektes Rechteck
Perfektes Rechteck
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🖼️ Images & Media (12)
File:Symmetries_of_square.svg
Symmetries_of_square.svg
File:PerimeterRectangle.svg
PerimeterRectangle.svg
File:Illustration for the area of a rectangle.svg
Illustration for the area of a rectangle.svg
File:Crossed rectangles.png
Crossed rectangles.png
File:Saddle rectangle example.png
Saddle rectangle example.png
File:Stacked bond.png
Stacked bond.png
File:Wallpaper group-cmm-1.jpg
Wallpaper group-cmm-1.jpg
File:Wallpaper group-p4g-1.jpg
Wallpaper group-p4g-1.jpg
File:Basketweave bond.svg
Basketweave bond.svg
File:Herringbone bond.svg
Herringbone bond.svg
File:Perfektes_Rechteck.svg
Perfektes_Rechteck.svg
File:smallest_perfect_squared_squares.svg
smallest_perfect_squared_squares.svg
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