Q:

# How many planes of symmetry does a cube have?

A:

A cube has nine planes of symmetry. Three of the planes run parallel to the faces of the cube, and the other six run diagonally from one edge to the opposite edge.

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Credit: Jared Tarbell CC-BY 2.0

If the cube has side length "s" then the three planes of symmetry that run parallel to the cube's faces form squares with side length "s" and none of the original vertices of the original cube. The other six planes form rectangles with length "s" and width "s*sqrt(2)" and all verticies part of the original cube.

Cutting the cube along the square planes of symmetry forms a rectangular prism with length and width "s" and height "s/2." Cutting the cube along the rectangular planes of symmetry forms a triangular prism with two triangular faces with side length "s" and height "(s/2)*sqrt(2)," two square faces with side length "s" and a base with side length "s" and width "s*sqrt(2)"

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