Answer:
The polygon has 9 sides (a regular nonagon).
Explanation:
Angles in a Regular Polygon
A polygon with n sides has a total sum of internal angles equal to 180°(n-2). This means that each angle (in a regular polygon) measures

The regular polygon used by Dina for the logo is such that each interior angle measures 140°, thus:

Multiply by n:

Operate:

Reducing:

Solving:
n=9
The polygon has 9 sides (a regular nonagon).