What is the product of the number of vertices on a cube and the number of degrees in each angle of an equilateral triangle?
Answers
Answer:
The reason the shape changes from 3D to 2D when you get to six triangles has to do with the number of degrees in an equilateral triangle. There are 60 degrees at each vertex of the triangle, so when you have 3 triangles connecting, like in the tetrahedron, there is a total of 180 degrees at that point. For an octahedron, the angles of the 4 triangles add up to equal 240 degrees, and for an icosahedron there are 300 degrees where the 5 triangles meet at each vertex. For each of these less-than-360-degree angles you get positive curvature and the shapes bend around into regular three-dimensional polyhedrons—but when you have 6 triangles connecting at a vertex, the angle reaches 360 degrees (a circle) and the shape flattens into a two-dimensional plane.
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