Which of these shapes would never tessellate? A. Hexagons B. Equilateral triangles C. Octagons D. Squares

Equilateral triangles tesselate the plane because the interior angle 60 degrees is a factor of 360. We're able to tile these triangles so that there are no gaps or overlaps, and we can cover the plane. Think of it like tiling a wall. So we rule out choice B.

Combining 6 equilateral triangles together forms a regular hexagon, so we can rule out choice A as well (I'm assuming your teacher meant to say "regular" in front of "hexagon"; as some non-regular hexagons do not tessellate).

As I mentioned with the wall tiling example, squares also tesselate the plane. Simply look at any floor tile or wall tile for a real world application. Another application is any xy grid. The 90 degree interior angle is a factor of 360. Therefore, we rule out choice D.

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In the previous section, we ruled out choices A, B, and D.

So the only thing left are octagons. The interior angle is 180(n-2)/n = 180(8-2)/8 = 135 degrees which is not a factor of 360. Note how 360/135 = 2.667 approximately which isn't a whole number.

So if we tried to glue octagons together such that there are no gaps or overlaps, then we'll find that it's impossible to do.

If you wanted octagons to be part of a tessellation, then you would need to involve squares as well. Otherwise there will be gaps.

This is why octagons do not tesselate the plane.