A pentagon puzzle

The shapes will

Proof

Image

It will

This kind of generalises.

I will try to provide as simple a solution as possible. Hopefully no pictures will be necessary.

It is well known that

Therefore,

There are 180 degrees * (n-sides minus 2) in an n-sided polygon. For a pentagon, each angle is therefore 540 degrees / 5, or 108 degrees

The amount of change in angle for a pair of adjacent pentagons will be (108 + 108) - 180, or 36 degrees.

Take 360 degrees in a circle and it is evenly divisible by 36 degrees, so the completed series of pentagons will consist of 10 pentagons and they will perfectly join up.

The answer is

And I'm going to (at least try to) proof that this generalizes to all regular polygons:

Examples

Square Heptagon Octagon