Polygon Diagonals Calculator

About this calculator

A diagonal is any line segment connecting two non-adjacent vertices of a polygon. For a polygon with n sides, the number of diagonals is given by the formula n(n − 3)/2 — each of the n vertices can connect to n − 3 others (excluding itself and its two adjacent neighbors), and dividing by 2 corrects for counting each diagonal twice. The sum of a polygon's interior angles follows a separate but related formula, (n − 2) × 180°, which comes from the fact that any n-sided polygon can be split into n − 2 triangles by drawing diagonals from a single vertex.

These formulas are foundational in geometry education and appear constantly in computational geometry, where counting diagonals relates to triangulation algorithms used in computer graphics, mesh generation, and GIS (geographic information systems) for breaking complex polygons into triangles for rendering or area calculation. Architects and structural engineers also reference diagonal counts when analyzing bracing patterns in polygonal trusses and frameworks.

This calculator takes the number of sides of a polygon and returns both its diagonal count and its interior-angle sum.