Step by step
The question
Question: what is the order of rotational symmetry of a regular octagon?
1 Mark a corner
Put a dot on one corner, so we can follow it.
Then turn the shape about its center.
2 Turn and count
After a turn of 45°, it fits on itself again.
That is 1.
After a quarter turn, it fits on itself again.
That is 2.
After a turn of 135°, it fits on itself again.
That is 3.
After a half turn, it fits on itself again.
That is 4.
After a turn of 225°, it fits on itself again.
That is 5.
After three quarter turns, it fits on itself again.
That is 6.
After a turn of 315°, it fits on itself again.
That is 7.
One more turn: the dot is back at the start.
That is 8.
3 The order
It fits 8 times in one full turn.
So the order is 8.
Careful!
The start and the full turn are the same place.
Count it once.
4 Check
Let's check.
One full turn is 360 degrees.
Shared by 8, each turn is 45°.
8 turns of 45 degrees make 360 degrees: one full turn.
Right!
Let's look back.
First: mark a corner.
Then: turn and count.
Then: the order.
Last: check.