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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 and width , then: [11] it has area =; it has perimeter = + = (+); each diagonal has length = +; and
Given a rectangle of height 1, length and diagonal length . The triangles on the diagonal have altitudes 1 / ς − 1 ; {\displaystyle 1/{\sqrt {\varsigma -1}}\,;} each perpendicular foot divides the diagonal in ratio ς 2 {\displaystyle \varsigma ^{2}} .
Assume a golden rectangle has been constructed as indicated above, with height 1, length and diagonal length +. The triangles on the diagonal have altitudes 1 / 1 + φ − 2 ; {\displaystyle 1/{\sqrt {1+\varphi ^{-2}}}\,;} each perpendicular foot divides the diagonal in ratio φ 2 . {\displaystyle \varphi ^{2}.}
Assume a silver rectangle has been constructed as indicated above, with height 1, length and diagonal length +. The triangles on the diagonal have altitudes 1 / 1 + σ − 2 ; {\displaystyle 1/{\sqrt {1+\sigma ^{-2}}}\,;} each perpendicular foot divides the diagonal in ratio σ 2 . {\displaystyle \sigma ^{2}.}
A root-phi rectangle divides into a pair of Kepler triangles (right triangles with edge lengths in geometric progression). The root-φ rectangle is a dynamic rectangle but not a root rectangle. Its diagonal equals φ times the length of the shorter side. If a root-φ rectangle is divided by a diagonal, the result is two congruent Kepler triangles.
where = + is the length of the rectangle's diagonal. If the two points are instead chosen to be on different sides of the square, the average distance is given by [ 3 ] [ 4 ] ( 2 + 2 + 5 ln ( 1 + 2 ) 9 ) s ≈ 0.869009 … s . {\displaystyle \left({\frac {2+{\sqrt {2}}+5\ln(1+{\sqrt {2}})}{9}}\right)s\approx 0.869009\ldots s.}
The diagonals of a square are (about 1.414) times the length of a side of the square. This value, known as the square root of 2 or Pythagoras' constant, [1] was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles.
A supergolden rectangle is a rectangle whose side lengths are in a : ratio. Compared to the golden rectangle , the supergolden rectangle has one more degree of self-similarity . Given a rectangle of height 1 , length ψ {\displaystyle \psi } and diagonal length ψ 3 {\displaystyle {\sqrt {\psi ^{3}}}} (according to 1 + ψ 2 = ψ ...