When you look through a kaleidoscope or simply at a small image reflected in
two mirrors touching long one side, you might notice that sometimes there are a countable number of images around a single point. Sometimes there seem to be partial images. This is easier to see with a simple two mirror kaleidoscope.
If you move the mirrors until you see exactly 6 clear images around the center, I can tell you that the angle between the two mirrors is 60 degrees.
If you move the mirrors until there are 9 clear images, the angle is 40 degrees.
If you move the mirrors until there are 10 clear images, the angle is 36 degrees.
You should now be able to tell me the angles of the mirrors when you see 15 clear images, 18 clear images, or 36 clear images.
You see 15 clear images when the angle is 24 degrees. You see 18 clear images when the angle is 20 degrees. You see 36 clear images when the angle is 10 degrees.
I assume that you know the number of degrees in a circle. Each time you look through the kaleidoscope and focus on a certain point, you can count the number of images that form a full circle around that point.