Diagonal

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Polygons

A diagonal of a polygon is any segment joining two vertices other than an edge.

Triangles have no diagonals while convex quadrilaterals have two interior diagonals and concave quadrilaterals have on interior and one exterior diagonal. The number of edges plus the number of diagonals of a polygon with n vertices is equal to $n \choose 2$

Polyhedra

Polyhedra have two different kinds of diagonals, face diagonals and space diagonals. A face diagonal of a polyhedron is a diagonal of one of the faces of the polyhedron, while a space diagonal is any segment joining two vertices which is neither an edge nor a face diagonal.

Tetrahedra have no space or face diagonals. Octahedra have no face diagonals but have 3 space diagonals. Cubes have 12 face diagonals (2 on each face) and 4 space diagonals. The number of edges plus the number of face diagonals plus the number of space diagonals of a polyhedron with n vertices is equal to $n \choose 2$.

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