# Resources tagged with: Golden ratio

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### There are 20 results

Broad Topics > Fractions, Decimals, Percentages, Ratio and Proportion > Golden ratio

### Golden Thoughts

##### Age 14 to 16 Challenge Level:

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.

### Golden Mathematics

##### Age 16 to 18

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

### Pentakite

##### Age 14 to 18 Challenge Level:

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

### Golden Powers

##### Age 16 to 18 Challenge Level:

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

### Golden Eggs

##### Age 16 to 18 Challenge Level:

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

### Golden Fibs

##### Age 16 to 18 Challenge Level:

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

### Golden Fractions

##### Age 16 to 18 Challenge Level:

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

### Pentabuild

##### Age 16 to 18 Challenge Level:

Explain how to construct a regular pentagon accurately using a straight edge and compass.

### Darts and Kites

##### Age 14 to 16 Challenge Level:

Explore the geometry of these dart and kite shapes!

### Leonardo of Pisa and the Golden Rectangle

##### Age 7 to 16

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

### Golden Triangle

##### Age 16 to 18 Challenge Level:

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.

### Pent

##### Age 14 to 18 Challenge Level:

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.

### Golden Ratio

##### Age 16 to 18 Challenge Level:

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

##### Age 16 to 18

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?

### Golden Construction

##### Age 16 to 18 Challenge Level:

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

### The Golden Ratio, Fibonacci Numbers and Continued Fractions.

##### Age 14 to 16

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.

### Whirling Fibonacci Squares

##### Age 11 to 16

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

### Gold Again

##### Age 16 to 18 Challenge Level:

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

### Pythagorean Golden Means

##### Age 16 to 18 Challenge Level:

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.

### Gold Yet Again

##### Age 16 to 18 Challenge Level:

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