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There are **16** NRICH Mathematical resources connected to **Golden ratio**, you may find related items under Fractions, decimals, percentages, ratio and proportion.

Problem
Primary curriculum
Secondary curriculum
### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Gold Yet Again

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Whirling Fibonacci Squares

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.

Age 11 to 16

Article
Primary curriculum
Secondary curriculum
### The Golden Ratio, Fibonacci Numbers and Continued Fractions.

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

Age 14 to 16

Problem
Primary curriculum
Secondary curriculum
### Golden Fractions

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Leonardo of Pisa and the Golden Rectangle

Leonardo who?! Well, Leonardo is better known as Fibonacci and this article will tell you some of fascinating things about his famous sequence.

Age 7 to 16

Problem
Primary curriculum
Secondary curriculum
### Golden Fibs

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pentakite

Given a regular pentagon, can you find the distance between two non-adjacent vertices?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Golden Ratio

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### About Pythagorean Golden Means

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Pent

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Golden Powers

You add 1 to the golden ratio to get its square. How do you find higher powers?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Golden Triangle

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagorean Golden Means

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Gold Again

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

Age 16 to 18

Challenge Level