Golden Rectangle Calculator

Created by Mateusz Mucha and Piotr Małek
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Jul 26, 2022

The golden rectangle calculator will calculate the length of either side and the area of the golden rectangle, provided you give the other side. Before you use this calculator, you should understand what a golden rectangle is, how to calculate ratios in general, and the formula for the golden ratio.

What is the golden rectangle?

The golden rectangle is a rectangle whose sides are in the golden ratio, that is (a + b)/a = a/b = φ, where a is the width, a + b is the length of the rectangle, and φ is the golden ratio: φ = (1+√5)/2.

Golden rectangle, marked with lengths a, b, and a+b

The ratio calculator is an effective tool to assist in calculating ratios in general, while the golden ratio calculator will do the same as the golden rectangle calculator, with the exception of finding the area of the rectangle.

How to use our golden rectangle calculator?

Here are the steps:

  1. Enter the width a.
  2. Enter the length a + b or segment b
  3. Find the area a × (a + b)
  4. If you know the area, divide by the missing part to get the other part.
  5. Check your answer with the golden rectangle calculator.

An interesting aspect of the golden rectangle is that when the square section is removed, the remainder is another golden rectangle. Also, adding another square to the rectangle with a side length of a + b is another golden rectangle. The golden rectangle calculator will verify this.

How do I draw a golden rectangle?

To construct a golden rectangle with a compass and ruler:

  1. Construct a square. Its side will become the width of a golden triangle.
  2. By constructing the perpendicular bisector, find the midpoint of one side of your square.
  3. Draw an arc with the center at this midpoint and a radius equal to the distance between this midpoint and either of the opposite vertices of the square.
  4. The intersection point of the arc and the line at which the compass needle sits gives you the desired vertex of the golden rectangle.

FAQ

How do I find the width of a golden rectangle?

To compute the width of a golden triangle given its length, divide the length by the golden ratio (1 + √5)/2, that is, approximately, by 1.618.

What is the width of a golden rectangle that is 32 cm long?

Approximately 19.777 cm. This is because the ratio of length to width is equal to (1 + √5)/2 ≈ 1.618; hence width ≈ 32cm / 1.618 ≈ 19.777

Who discovered the golden rectangle?

The golden rectangle was undoubtfully known to the Ancient Greeks. According to some historians, the Babylonians were the first to discover this concept.

Mateusz Mucha and Piotr Małek
Golden rectangle, marked with lengths a, b, and a+b
a
in
b
in
a + b
in
Area
in²
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