How many regular polygons have integer values for their interior angles?

Answer 1

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 sides

So 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 factors

Of 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.

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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 triangle
• 4 sides - square
• 5 sides - pentagon
• 6 sides - hexagon
• 8 sides - octagon
• 9 sides - nonagon
• 10 sides - decagon
• 12 sides - duodecagon
• 15, 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).

Answer 2

There are 22.