The diagram shows a nine-point pegboard. We'll take the board radius as one (in other words it's a unit circle).
two isosceles triangles on a pegboard
Looking at triangle OAM, why is AM equal to sin40° ?

Looking at triangle OAB, why is AB equal to 2sin20° ?

Looking at triangle ABM, why is AM also equal to ABcos20° which works out as 2sin20° cos20°

So it looks like the same AM length value can be calculated by using sin40°

and also calculated by using 2sin20°cos20°

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Now switch to a 10 point pegboard and find the two ways to calculate the AM length on that board.

How about on a 12 point board ? What general result is emerging ?
Draw a diagram and use it to provide reasoning which accounts for that general result.


Many thanks to Geoff Faux who introduced us to the merits of the 9 pin circular geo-board.
The boards, moulded in crystal clear ABS that can be used on an OHP (185 cm in diameter), together with a teacher's guide, are available from Geoff at Education Initiatives