Rectangle.html |
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In geometry, a rectangle is a quadrilateral where all four of its angles are right angles. It follows that a rectangle has two pairs of parallel sides; that is, a rectangle is a parallelogram. It is also a special case of a trapezoid (in North America) or trapezium (in Britain and elsewhere). A rectangle with vertices ABCD would be denoted as 
ABCD.
The three-dimensional counterpart of a rectangle is a cuboid, also called a rectangular parallelepiped.
A perfect rectangle may refer to the golden rectangle, or to a rectangle partitioned into regions, such as squares or right triangles, which are of the same shape but all of different sizes.
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Definitions and properties
An equilateral rectangle is known as a square. A non-square rectangle is known as an oblong, but the term "oblong" is generally not used today by mathematicians.
Normally, of the two opposite pairs of sides in a rectangle, the length of the longer side is called the length of the rectangle, and the length of the shorter side is called the width or the breadth.
The area of a rectangle is the product of its length and its width; in symbols, A = lw. For example, the area of a rectangle with a length of 5 and a width of 4 would be 20, because 
.
The ratio of the longer side to the shorter side of a rectangle is called the aspect ratio or eccentricity of the rectangle.
In a rectangle the diagonals cross each other at their respective midpoints, under the same argument as for parallelograms. Unlike general parallelograms the two diagonals of a rectangle have the same length, the length of the diagonal can be found using the Pythagorean theorem.
In calculus, the Riemann integral can be thought of as a limit of sums of the areas of arbitrarily thin rectangles.
In non-Euclidean geometry, rectangles with right angles do not exist.
Perfect rectangle
The golden rectangle has been called the perfect rectangle in the belief that it is the most aesthetically pleasing, though experiments have "yielded only the cloudy conclusion that most people prefer a rectangle somewhere between a square and a rectangle that is twice as long as it is wide"1.
A perfect rectangle is also a rectangle R tiled by a finite number n of similar tiles, no two of which are the same size2 3 4. n is the order of the tiling. If there are tiles of the same size, R is an imperfect rectangle.
It is unnecessary to call a tiling imperfect when only congruent tiles are being considered, copies of a single polyomino for example.
Sprague5 and Brooks et al.2 proved that every rectangle with commensurable sides is perfectible with squares in infinitely many ways. The lowest-order perfect squared rectangle has 9 squares2, and the lowest-order perfect squared square has 21 squares.
A perfect rectangle tiled by isosceles (non-isosceles) right triangles has a minimum of 5 (3) tiles.
See also
External links
- Eric W. Weisstein, Rectangle at MathWorld.
- Definition and properties of a rectangle With interactive animation
- Area of a rectangle with interactive animation
References
- ^ M. Gardner (1966). "More Mathematical Puzzles and Diversions", Pelican Books p.76 ISBN 0140207481
- ^ a b c R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte (1940). "The dissection of rectangles into squares". Duke Math. J. 7 (1): 312–340. doi:, http://projecteuclid.org/euclid.dmj/1077492259.
- ^ J.D. Skinner II, C.A.B. Smith and W.T. Tutte (November 2000). "On the Dissection of Rectangles into Right-Angled Isosceles Triangles". J. Combinatorial Theory Series B 80 (2): 277–319. doi:.
- ^ Eric W. Weisstein. PerfectRectangle at MathWorld, Invalid definitions, http://mathworld.wolfram.com/PerfectRectangle.html. Retrieved on 16 October 2008.
- ^ R. Sprague (1940). "Ũber die Zerlegung von Rechtecken in lauter verschiedene Quadrate". J. fũr die reine und angewandte Mathematik 182: 60–64.