Andrea's assumption is correct: the three tiles won't fit together perfectly because sum of the interior angles is not 360 deg
How to show the diagram will not work
The proof will be achieved using interior angles of regular polygons. This can be calculated using the formula:
Interior Angle = (n - 2) * 180 / n
where n is the number of sides.
Solving using n for various cases
An octagon (n = 8) has interior angles of 135
A pentagon (n = 5) has interior angles of 108
A hexagon (n = 6) has interior angles of 120
Angle at a point is 360, therefore adding the angles suppose ot be 360
135 + 108 + 120 = 363
Since \363 ≠ 360 Andrea's assumption is correct: the three tiles won't fit together perfectly