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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?

Cosines Rule

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Coke Machine

Age 14 to 16 Challenge Level:

This is another tough nut and perhaps the diagram of the 50p piece will help.

Fifty P
A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing the straight line DE with an arc centred at A and radius AE; replacing the straight line EF with an arc centred at B radius BF ...etc..

The 50p piece can roll in the same chute as a disc of radius $r$. Suppose the seven arcs forming the edge of the 50p piece (the arcs AB, BC etc. ) all have radius $R$ (where $R$=AD=AE=BE=BF...) then you need to find $R$ in terms of $r$. These seven arcs subtend angles of $2\pi /7$ at the centre of the disc and $2\pi /14$ at the opposite edge.