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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?

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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.

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Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.


Stage: 4 and 5 Challenge Level: Challenge Level:1

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