Hexapentagon
Weekly Problem 53 - 2007
The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of x?
The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of x?
Age
11 to 14
Challenge level
Problem
Image
The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of $x$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
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 $x=84$.