How many triangles in a cube?

Answer 1

56

Triangles

Answer 2

I suppose the corner points (all 8 points) can be connected inside the cube. According to Miller indices, there are three possible face families, that is, {1 0 0}, {1 1 0} and {1 1 1}. Because there are 4 different triangles in a square (or a rectangle), we have 24 triangles on the surfaces of a cube. That is, 4 times 6, where 6 means six {1 0 0} faces. Similarly, on the internal {1 1 0} faces, there are 4 triangles on each six face. This gives another 24 possible triangles. Finally, there are eight {1 1 1} triangles. These last triangle families can be easily visualized as follows. Suppose we draw (1 1 1) triangle in a cube. Next, rotate it by z-axis. That is, 90 degrees rotation gives second triangle. Another 90 degrees rotation gives third triangle. And finally, third rotation gives fourth triangle. This makes double cross on the bottom surface.

The maximum triangular side on the bottom surface is four. Therefore, another set of top surface triangles with double cross of triangular sides yields 4 triangles. When double cross is drawn on bottom surface, single cross is made on each four side surface. Similarly, when the top surface double cross is made another single cross is made on the side surfaces. Therefore, each 8 triangle doesn't overlap. Therefore, there are 56 triangles in a cube.