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Which regular polygons tile around a point

Which regular polygons tile around a point

IP 1
6 marks
  1. Show that each interior angle of a regular \(n\)-sided polygon is \(\left(180-\dfrac{360}{n}\right)^\circ\).[1]
  2. Copies of one regular polygon are to fit together around a point with no gaps or overlaps. Show that \(k=\dfrac{2n}{n-2}\) copies are needed, and hence explain why the only possible polygons are the equilateral triangle, the square and the regular hexagon.[3]
  3. A square, a regular hexagon and a regular \(n\)-sided polygon fit together around a point with no gaps. Find \(n\).[2]

Solution:

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Answer:(a) exterior angle \(\dfrac{360^\circ}{n}\), so interior angle \(180^\circ-\dfrac{360^\circ}{n}\) (b) \(k=\dfrac{2n}{n-2}=2+\dfrac4{n-2}\) is whole only for \(n=3,4,6\) (c) \(n=12\)

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