Let the number of sides be n and the angle not included in the
sum of 2270° have measure k°.

Then:

 (n-1)180° = 2270° + k°  

180n - 180 = 2270 + k

      180n = 2450 + k

         n = 

Since n must be a positive integer, 2450 + k must be a multiple of 180.

The smallest multiple of 180 greater than 2450 is 2520 or 14*180 which would 

make k = 2520-2450 or 70.  Thus n=14 (with the omitted angle 70°) is a 
solution.

The next multiple of 180°, 2700 or 15×180 would cause k° to be 250° which
would give us a concave polygon.  But we are given that the polygon is
convex, i.e., all interior angles are less than 180°.

Therefore there is just one solution, a polygon of 14 sides.

Edwin