Review question

# How can we arrange regular polygons around a point? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6908

## Question

The sum of the interior angles of a polygon with $n$ sides is $(2n-4)$ right angles. Deduce that each interior angle of a regular polygon is $\left( 2 - \dfrac{4}{n} \right)$ right angles.

Three regular polygons, two having $n$ sides and one having $p$ sides, fit together exactly at a common vertex, as shown:

By using the fact that the sum of the three angles shown is four right angles, prove that $\frac{4}{n} + \frac{2}{p} = 1.$

Show that this formula can be rearranged into the form $p = \frac{2n}{n-4}.$ By inserting into this formula various values of $n$, list all the pairs of positive integers $(n,p)$ for which $n < 12$ which satisfy this equation. By means of a sketch, interpret one of these solutions where $n$ and $p$ are different in terms of three regular polygons meeting at a common vertex.