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# Six Discs

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### Coke Machine

### At a Glance

### Cosines Rule

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Age 14 to 16

Challenge Level

Six circular discs, all of radius 1 unit, are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box.

- Without using any calculations, compare the areas of the bases of boxes A and B and say which is the larger.
- Use estimation to list the packings in order of increasing base area. Which are the hardest to put in order?
- Now calculate the area of the base of each box and compare the correct order with your estimate.

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

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

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.