Regular Polygon Diagonal Calculator

Number of Diagonals:

Diagonal Length:

Interior Angle:

Understanding Regular Polygons

Basic Properties

Number of Diagonals = n(n-3)/2

Interior Angle = (n-2)×180°/n

Exterior Angle = 360°/n

Symmetry

Regular polygons have rotational and reflective symmetry

Angles

All interior angles are equal

Sides

All sides have equal length

Diagonal Properties

Diagonal Length = 2R×sin(kπ/n)

where R is circumradius and k is diagonal order

Types of Diagonals

Different lengths based on vertices connected

Shortest Diagonal

Connects nearest non-adjacent vertices

Longest Diagonal

Passes through center for odd n

Advanced Properties

Circle Relations

Inscribed and circumscribed circles

Area Properties

Area divided by diagonals

Symmetry Groups

Dihedral group Dn

Special Cases and Applications

Regular Polygons in Nature

  • Crystal Structures
  • Molecular Geometry
  • Biological Forms
  • Honeycomb Patterns

Architectural Applications

  • Building Design
  • Floor Plans
  • Structural Elements
  • Decorative Patterns

Mathematical Connections

  • Trigonometry
  • Group Theory
  • Tessellations
  • Golden Ratio