Integral polygons
Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have?
Age
11 to 14
Challenge level
Problem
Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees.
What is the greatest number of sides the polygon could have?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
The greatest number of sides the polygon could have is $360$.
As each interior angle of the polygon is a whole number of degrees, the same must apply to each exterior angle. The sum of the exterior angles of a polygon is $360^{\circ}$ and so the greatest number of sides will be that of $360$-sided polygon in which each interior angle is $179^{\circ}$, thus making each exterior angle $1^{\circ}$.
As each interior angle of the polygon is a whole number of degrees, the same must apply to each exterior angle. The sum of the exterior angles of a polygon is $360^{\circ}$ and so the greatest number of sides will be that of $360$-sided polygon in which each interior angle is $179^{\circ}$, thus making each exterior angle $1^{\circ}$.