- When $n$ is 180 or 360, the angle is a whole number. It seems to be going up by 180.
- When $n$ is a multiple of $7$, the angle is a decimal.
- When $n$ is any of $150, 250, 350, \ldots$ the angle is a decimal
- If both $n$ and $n-2$ are prime (except $n=5$!), then the angle is a decimal
- If $n$ is a factor of $12$, then the angle is a whole number
- Some multiples of $12$ give whole number angles
- If $n$ is prime, greater than $10$, and not a multiple of $6$, then the angle is a decimal
- If $n=12345678912$, then the angle is a whole number
The first four regular polygons have whole number angle measurements, but a heptagon has a decimal 128.57 angle measurement. Why did the whole numbers go away?
My students are learning to become Independent Problem Solvers and one approach they use when solving a problem is Trying Multiple Examples. Students calculated the interior angle measurements for several more regular polygons and made observations. (I've reworded them slightly.)
$n$ is the number of sides of a regular polygon.
Some of these observations are incorrect, and none of them exactly capture when the angle is a whole number. However, they all contain the germ of an idea. I was impressed to see my students willing to make a hypothesis, even if it was incomplete. This is the heart of being an Independent Problem Solver.
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Philadelphia, Pennsylvania. Your comments are always welcome.