If a 12-sided regular polygon rotates about its center, at which angle of rotation will the image of the polygon coincide with the preimage?

Question

75°



45°



36°



30°

Answer 1

At the center of the polygon are twelve 30° angles - 360° / 12.

If the polygon is rotated 30° the image (transformation) shall coincide with the preimage (the figure before transformation).

Answer 2

As David W pointed out,

A 12 sided regular polygon divides 360 degrees into 12 equal segments... 360/12 = 30

Thus each equal section is 30 degrees.

Any point where the polygon coincides with the preimage will be a multiple of 30 degrees.

From your list the only choice is 30 degrees.

Answer 3

In the 1960’s, Seymour Papert introduced Turtle Graphics (now available in several computer programming languages).  The robot turtle starts in a direction, turns left a certain amount, moves forward, turns left the same amount again, moves forward, .. and when it reaches the starting point, turns left for one last time.  The sum of the left turns equals 360 degrees (the turtle has completely rotated around).  In math terms, the sum of the external angles of a regular polygon is 360 degrees.

Here, we have a 12-sided regular polygon which rotates until the image coincides with the preimage.  That is 360/12 = 30 degrees (the outside angle).  If it rotated thus 12 times, it would have made one complete rotation.