# Polygon Cradle

Weekly Problem 18 - 2007

A regular pentagon together with three sides of a regular hexagon form a cradle. What is the size of one of the angles?

A regular pentagon together with three sides of a regular hexagon form a cradle. What is the size of one of the angles?

## Problem

Image

The figure shows a regular pentagon $PQRST$ together with three sides $XP$, $PR$, $RU$ of a regular hexagon with vertices $PRUVWX$. What is the size of angle $SRU$?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

$48^{\circ}$

As $PQRST$ is a regular pentagon, each of its internal angles is $108^{\circ}$. The internal angles of the quadrilateral $PRST$ add up to $360^{\circ}$ and so by symmetry $\angle PRS=\angle RPT=\frac{1}{2}(360^{\circ}-2\times 108 ^{\circ})=72^{\circ}$. Each interior angle of a regular hexagon is $120^{\circ}$, so $\angle PRU=120^{\circ}$.

Therefore $\angle SRU=\angle PRU-\angle PRS=120^{\circ}-72^{\circ}=48^{\circ}$.