3 Answers:
- Equilateral triangles
- Squares
- Regular Hexagons
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Reason:
The interior angle formula of a regular polygon is
i = 180*(n-2)/n
where n = number of sides, and i = interior angle in degrees.
If n = 3, then each interior angle would be i = 60. Note how this interior angle is a factor of 360. This explains why equilateral triangles are a type of regular polygon that tessellates the plane.
If n = 4, then i = 90 which is also a factor of 360. This means squares are another type of regular polygon that tessellate the plane.
Unfortunately n = 5 leads to i = 108 which is not a factor of 360; therefore, regular pentagons do not tessellate the plane.
Luckily, n = 6 works because i = 120 is a factor of 360.
Any larger value of n will lead to some value of i that isn't a multiple of 360. Therefore, only equilateral triangles, squares, and regular hexagons are the only regular polygons that tessellate the plane.