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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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.

Darts and Kites

Explore the geometry of these dart and kite shapes!


Age 14 to 18
Challenge Level

Why do this problem?

The diagram is not given because it is a good learning experiene for students to draw their own diagrams. The problem calls for simple geometrical reasoning involving angles and similar triangles. It is a good way in to the wonderful mathematics related to the golden ratio and the Fibonnaci sequence.

Possible approach

Offer a selection of examples on 'Golden Mathematics' (see links below).

Key questions

What do you know about the angles of a reguar pentagon?

Can you spot any isosceles triangles?

Can you spot any similar triangles? ... any congruent triangles?

Can you use the geometry to give you an equation involving the unknown you have to find?

Possible support

Golden Trail