Copyright © University of Cambridge. All rights reserved.

'Hexapentagon' printed from https://nrich.maths.org/

Show menu


Each interior angle of a regular pentagon is $108$ degrees, whilst each interior angle of a regular hexagon is $120$ degrees. The non-regular pentagon in the centre of the diagram contains two angles which are interior angles of the regular hexagon, two angles which are interior angles of the regular pentagon and a fifth angle, the one marked $x$. So: $x+2\times120+2\times108= 5\times108=540$. Hence $x=84$.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.