Question

Which regular polygon has a minimum rotation of 45degree to carry the polygon onto itself?

Answer 1

Everything up to an octagon.

Explanation:

Imagine a circumscibed circle around the polygon as a wheel, the spokes being the lines connecting the corners of the polygon to the centre of the circle.
Now if we turn the wheel, every time a spoke comes to the position of the previous spoke, we 'carry the polygon onto itself'.

A triangle or 3-gon has three 'spokes' so in a full turn this happens three times, or every 1/3 of a turn of 360 degrees = every 120 degrees.

This goes for all n-gons, so the required rotation is `360^o/n`

Now we have to find out for which `n` the rotation is`>=45^o`:

`360/n>=45->360>=45n->n<=360/45->n<=8`