diagonals

Diagonals are line segments joining two nonadjacent vertices of a polygon. In a broader sense, the term also applies to polyhedra: a diagonal is a segment joining two vertices that are not connected by an edge, which may lie inside the solid (space diagonals) or on a face (face diagonals).

For a simple n-gon, the number of diagonals is n(n-3)/2. This counts each diagonal twice, once from each endpoint.

In convex polygons, all diagonals lie inside the polygon. In concave polygons, some diagonals can lie outside the boundary of the polygon, depending on the positions of the vertices.

Quadrilaterals have exactly two diagonals; these diagonals intersect each other inside the figure. In parallelograms, the diagonals bisect each other, meaning they cut each other into segments of equal length.

In regular polygons, diagonals come in several distinct lengths, determined by how many vertices they skip. If the polygon is inscribed in a circle of radius R, a diagonal that spans k edges has length D_k = 2R sin(kπ/n) for 2 ≤ k ≤ floor(n/2). When k = n/2 in even polygons, the diagonal is a diameter across the circle.

In prisms and other polyhedra, diagonals are further categorized as space diagonals (through the interior) and face diagonals (within a face). Diagonals are foundational in topics such as polygon triangulation in computer graphics and geometric reasoning in mathematics.

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