Triangle Tilings: Nice Angles We use the term nice angle to refer to angles which are 180/n degrees for some whole number n. So the nice angles are (in degrees) 180/2 = 90, 180/3 = 60, 180/4 = 45, 180/5 = 36, 180/6 = 30, 180/7 = 25.714..., 180/8 = 22.5, 180/9 = 20, and so on (there are infinitely many). It turns out that in order for a triangle to fit evenly into a tiling the angle at each of its corners must be a nice angle. This is so that an even number of the triangles will fit around that corner. For example, if a triangle has a 90 degree angle, then 4 copies of that triangle will exactly fit around that corner.

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