Interior angles in regular polygons

Equilateral triangles have interior angles of 60o​, squares 90o​ and pentagons 108o​. The first regular n-sided polygon that doesn't have a whole number ​as an interior angle is the heptagon.

When exploring interior angles with my students we'd usually deduce a rule and use it to find interior angles of up to 9, 10 or even 12 sided shapes (with an occasional 100 sided polygon thrown in) before moving on.

However this week I was asked the question "how many regular polygons are there with integer interior angles?"

(image from MEP)

How would you find a solution? Fortunately, the question used some of the maths in last weeks post How many factors does 300 have?

You can find more ideas and activities like this in the joint STEM Leanring-NRICH lists below:

NRICH Angles, Polygons and Geometrical Proof - Stage 3

NRICH Angles, Polygons and Geometrical Proof - Stage 4

 

Subject(s)Mathematics
Age11-14, 14-16
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