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Triangles have no diagonals, while convex quadrilaterals have two interior diagonals, and concave quadrilaterals have one interior and one exterior diagonal. The number of diagonals in a polygon can be determined by the formula .
To prove this is quite simple. We have an n-gon and we choose one point. From this point, there are diagonals we can make form this point; the ones we don't count are itself and the two vertices that form sides with it. There are n vertices, so for each of these there are diagonals, thus, there are diagonals. However, we are not done. When counting our diagonals, we counted some twice; we used the same vertices, just backwards. (i.e. In quadrilateral ABCD, diagonals AC and CA are the same.) So, we simply divide by 2 to get our final formula, .
The number of edges plus the number of diagonals of a polygon with n vertices is equal to .
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 .