If the interior angle of a regular polygon is an integer, then so is the exterior angle.For a regular polygon, the exterior angles are all the same and can be calculated as 360° ÷ number of sides; for this to be an integer, the number of sides must be a factor of 360.To be a polygon, the shape must have at least 3 sidesSo the question is the same as how many factors are there of 360 which are greater than or equal to 3.360 = 2³ × 3² × 5 → it has (3+1)×(2+1)×(1+1) = 4×3×2 = 24 factorsOf these, the factors 1 (= 2⁰ × 3⁰ × 5⁰) and 2 (= 2¹ × 3⁰ × 5⁰) are less than 3, so there are 24 - 2 = 22 factors greater than or equal to 3.→ there are 22 regular polygons with integer interior angles.--------------------------------------The factors of 360 are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360→The regular polygons with integer interior angles are:3 sides - equilateral triangle4 sides - square5 sides - pentagon6 sides - hexagon8 sides - octagon9 sides - nonagon10 sides - decagon12 sides - duodecagon15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 sides (not really met, so you can look up the names if you are really interested).