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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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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.


Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?

The stimulus for the problem is the engaging context of the construction of the solid. Is this really a regular dodecahedron and how can we be sure?

Possible approach

Without discussing how or why the pentagon is regular, encourage learnersto make pentagons and put them together to make a dodecahedron.
Then challenge them to examine whether the pentagons are actually regular or not.

Key Questions

  • How do you know this is a regular pentagon?
  • What would have to be the case for the pentagon to be regular (all sides and all angles equal)?
  • What do we know?
  • What mathematics do you know that might be useful?


Making this and other solids in similar ways, see the article on constructing platonic solids. The focus for the learners is on reading and following written instructions as much as on gaining greater familiarity with 3D shapes.


The A4 paper part of theproblem.