# Special Rectangle

Definitions and formulas for the perimeter of a rectangle, the area of a rectangle, how to find the length of the diagonal of a rectangle, properties of the diagonals of a rectangle

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 perimeter of a rectangle area of a rectangle properties of the sidesand angles of a rectangle diagonal of a rectangle properties of the diagonalsof a rectangle
 The perimeter of a rectangle: To find the perimeter of a rectangle, just add up all the lengths of the sides:Perimeter= L + w + L + w = 2L + 2w
 The area of a rectangle: To find the area of a rectangle, just multiply the length times the width:Area= L x w

About Special Special is a sleek frame made from lustrous brown carbon fiber for a seriously polished look. Every bit stylish and understated, the sharp angles and subtle contours make this rectangular frame stand out from the rest. Designed with adjustable nose pads for a secure fit and spring hinges for added comfort. A rectangle is a quadrilateral with four right angles. Thus, all the angles in a rectangle are equal (360°/4 = 90°). Moreover, the opposite sides of a rectangle are parallel and equal, and diagonals bisect each other. Mystic Rectangle. The Mystic Rectangle, or Rectangle, is comprised of two opposition aspects, connected with two trine and two sextile aspects. The harmonious trine and sextile aspects can be considered release points for the tension of the oppositions. The native can be attracted to important causes. A rectangle is a two-dimensional plane figure with four sides. A rectangle is a four-sided polygon in which the opposite sides are parallel and equal to each other. It is one of the types of quadrilaterals in which all four angles are right angles or equal to 90 degrees. The rectangle is a special type of parallelogram with all its angles equal.

 The sides and angles of a rectangle: Opposite sides of a rectangle are the same length (congruent).The angles of a rectangle are all congruent (the same size and measure.)Remember that a 90 degree angle is called a 'right angle.' So, a rectangle has four right angles.Opposite angles of a rectangle are congruent.Opposite sides of a rectangle are parallel.
A golden rectangle with sides ab placed adjacent to a square with sides of length a produces a similar golden rectangle.

In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, ${displaystyle 1:{tfrac {1+{sqrt {5}}}{2}}}$, which is ${displaystyle 1:varphi }$ (the Greek letter phi), where ${displaystyle varphi }$ is approximately 1.618.

Golden rectangles exhibit a special form of self-similarity: All rectangles created by adding or removing a square are Golden rectangles as well.

A method to construct a golden rectangle. Owing to the Pythagorean theorem,[a] the diagonal dividing one half of a square equals the radius of a circle whose outermost point is also the corner of a golden rectangle added to the square.[1]

## Construction

A golden rectangle can be constructed with only a straightedge and compass in four simple steps:

1. Draw a simple square.
2. Draw a line from the midpoint of one side of the square to an opposite corner.
3. Use that line as the radius to draw an arc that defines the height of the rectangle.
4. Complete the golden rectangle.

A distinctive feature of this shape is that when a square section is added—or removed—the product is another golden rectangle, having the same aspect ratio as the first. Square addition or 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. Diagonal lines drawn between the first two orders of embedded golden rectangles will define the intersection point of the diagonals of all the embedded golden rectangles; Clifford A. Pickover referred to this point as 'the Eye of God'.[2]

## History

The proportions of the golden rectangle have been observed as early as the BabylonianTablet of Shamash (c. 888–855 BC),[3][4] though Mario Livio calls any knowledge of the golden ratio before the Ancient Greeks 'doubtful'.[5]

According to Livio, since the publication of Luca Pacioli's Divina proportione in 1509, 'the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use.'[6]

The 1927 Villa Stein designed by Le Corbusier, some of whose architecture utilizes the golden ratio, features dimensions that closely approximate golden rectangles.[7]

## Relation to regular polygons and polyhedra

Euclid gives an alternative construction of the golden rectangle using three polygons circumscribed by congruent circles: a regular decagon, hexagon, and pentagon. The respective lengths a, b, and c of the sides of these three polygons satisfy the equation a2 + b2 = c2, so line segments with these lengths form a right triangle (by the converse of the Pythagorean theorem). The ratio of the side length of the hexagon to the decagon is the golden ratio, so this triangle forms half of a golden rectangle.[8]

Three golden rectangles in an icosahedron

The convex hull of two opposite edges of a regular icosahedron forms a golden rectangle. The twelve vertices of the icosahedron can be decomposed in this way into three mutually-perpendicular golden rectangles, whose boundaries are linked in the pattern of the Borromean rings.[9]

## Notes

### Different Rectangle

1. ^${displaystyle {tfrac {1}{2}}^{2}+1^{2}={tfrac {5}{2^{2}}}}$

## References

1. ^Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. p. 11. ISBN9-781-61614-424-1.
2. ^Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York City: Broadway Books. p. 85. ISBN0-7679-0816-3.
3. ^Olsen, Scott (2006). The Golden Section: Nature's Greatest Secret. Glastonbury: Wooden Books. p. 3. ISBN978-1-904263-47-0.
4. ^Van Mersbergen, Audrey M., Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic, Communication Quarterly, Vol. 46, 1998 ('a 'Golden Rectangle' has a ratio of the length of its sides equal to 1:1.61803+. The Parthenon is of these dimensions.')
5. ^Livio, Mario. 'The Golden Ratio in Art: Drawing heavily from The Golden Ratio'(PDF). p. 6. Retrieved 11 September 2019.
6. ^Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York City: Broadway Books. p. 136. ISBN0-7679-0816-3.
7. ^Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. ISBN0-419-22780-6: 'Both the paintings and the architectural designs make use of the golden section'.
8. ^Euclid, Elements, Book XIII, Proposition 10.
9. ^Burger, Edward B.; Starbird, Michael P. (2005). The Heart of Mathematics: An Invitation to Effective Thinking. Springer. p. 382. ISBN9781931914413{{inconsistent citations}}CS1 maint: postscript (link).