### There are 20 results

Broad Topics >

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

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

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

##### 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.

##### 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!

##### 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.

##### 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

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

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

Explore the geometry of these dart and kite shapes!

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

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

##### 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.

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

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

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

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

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

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

##### 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?

##### 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.

##### 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.

##### 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.

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

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

##### Age 11 to 16

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

##### 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.

##### 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.