An octagon is a polygon with eight sides and eight vertices. In geometry the term covers both convex and concave eight‑sided figures, but most common discussion focuses on the regular octagon, where all sides and interior angles are equal. For a concise reference see the octagon entry.

Key characteristics

Important numeric and structural properties include:

  • Interior angle sum: (8−2)×180° = 1080° for any octagon.
  • Each interior angle of a regular octagon measures 135°; each exterior angle is 45°.
  • Diagonals: an octagon has 8(8−3)/2 = 20 diagonals in total.
  • Symmetry: a regular octagon has dihedral symmetry D8, of order 16 (eight rotations and eight reflections).

Formulas and construction

The area of a regular octagon with side length a is A = 2(1+√2) a². A regular octagon can be constructed geometrically by truncating the corners of a square or by placing eight equally spaced points on a circle (it is a cyclic polygon). Practical compass-and-straightedge constructions are standard in classical geometry.

History and cultural use

The name derives from Greek roots: octa- meaning "eight" and -gon meaning "angle". Octagonal forms appear in architecture (towers, baptisteries, pavilions) and decorative arts across many cultures because they offer a pleasing transition between square and circular plans. The octagon is also widely recognized in signage: the red stop sign in many countries uses a regular octagon for high visibility and symbolic recognition.

Applications and examples

Beyond signage and architecture, octagons are used in tiling patterns, emblem design, game boards, and in some coin shapes to aid handling. In mathematics and computer graphics, octagons serve as example polygons for studying symmetry, tiling, and polygonal approximation of circles.

Octagons may be convex or concave; a concave octagon has at least one interior angle greater than 180°. An octagram is a star polygon related to the octagon but formed by connecting alternate vertices. Do not confuse an octagon (2D) with an octahedron, which is a regular polyhedron with eight triangular faces.

For further geometric details and illustrations, consult the linked octagon resource.