Calculations at a golden rectangle, a rectangle, where the ratio of the two sides is the golden ratio. The golden ratio is defined by the equation (a+b)/a = a/b, it is slightly above 1.6. Enter one value and choose the number of decimal places. Then click Calculate.
Formulas:
b = a / φ
d = √ a² * ( 1 + 1/φ² )
p = 2a * ( 1 + 1/φ )
A = a² / φ
Golden ratio phi:
φ = ( 1 + √5 ) / 2 = 1.618033988749895...
Lengths and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
If you remove a square with the side length of the shorter side of the rectangle from a golden rectangle, you get another golden rectangle. If a line is divided in the golden ratio, two new lines are created with the property that the ratio of the longer to the shorter new line is exactly the same as the ratio of the original total line to the longer new line. The golden ratio can be found in the Fibonacci sequence, among other things, and has been used a lot in architecture and art because it is said to be particularly aesthetic. It was already known in greek antiquity, and its irrationality was proven by Euclid. However, it only became popular and used specifically in the Renaissance. The constant of the golden ratio, phi, is also known as the most irrational of all numbers, the one that is furthest away from its rational neighbors. Its continued fraction expansion contains only ones, it is 1 + ( 1 / ( 1 + ( 1 / ( 1 + (...) ) ) ) ).
If you connect two adjacent corners of a regular icosahedron and then form a rectangle to the two opposite corners, this is a golden rectangle. The square root of 5 often appears in the formulas for the icosahedron, as in the constant phi. The golden ratio can also be found in the pentagram, but there is no golden rectangle here.