How many degrees can a regular pentagon be rotated about its center so that it remains unchanged

Question

This is for marh, but the book does not explain it.

Answer 1

Rotational symmetry of a regular polygon is based on the number of vertices it has. If the figure is rotated a full 360 degrees, each vertex lands exactly on top of where it started. Since you have 5 vertices, your starting vertex can land in 5 different starting places and still come up with the same figure. So if you take 360/5 you get 72 degree rotations to each consecutive vertex.

 

This is saying that a rotation of 72, 144, 216,288 and 360 will result in the same figure. This is hard to explain without a visual, but I hope you get what I'm trying to say.

Answer 2

Assuming "remains unchanged" means the rotated pentagon
will look like the original figure position...

We can divide the pentagon (5 sides) into five inscribed triangles
with central vertex angles of 72 degrees.

360degrees ÷ 5 = 72degrees....complete rotation is 360 degrees

So each rotation of the pentagon of 72 degrees will be identical
to the figure in its initial position before rotation.

Answer 3

Assuming "remains unchanged" means the rotated pentagon 

will look like the original figure position...

 

We can divide the pentagon (5 sides) into five inscribed triangles

with central vertex angles of 72 degrees.  

 

360degrees ÷ 5 = 72 degrees....complete rotation is 360 degrees

 

So each rotation of the pentagon of 72 degrees will be identical

to the figure in its initial position before rotation.