Octagon
GeometryAn octagon is a polygon with eight sides and eight interior angles summing to 1080 degrees.
Formula
\text{Interior angle sum} = 1080^\circ; \text{each regular angle} = 135^\circ
Definition
An octagon is a polygon with eight sides and eight corners. The eight angles inside add up to $1080^\circ$. A regular octagon has all eight sides equal and each angle measuring $135^\circ$.
Example
A stop sign is a regular octagon. Some bathroom floor tiles and the UFC fighting cage are also octagonal. If you cut the corners off a square, you get a shape that looks like an octagon.
Key Insight
The prefix "octa-" means eight in Greek (like an octopus with eight arms). A regular octagon looks like a circle with the corners cut flat. Each interior angle is $135^\circ$ - halfway between $90^\circ$ (square) and $180^\circ$ (flat line).
Definition
An octagon has $8$ sides, $8$ vertices, interior angle sum $= (8-2) \times 180 = 1080^\circ$. A regular octagon has interior angles of $135^\circ$ and $20$ diagonals. A regular octagon can be formed by cutting equal isosceles right triangles from the corners of a square.
Example
Regular octagon with side $s$: area $= 2(1+\sqrt{2})s^2$. For $s=1$: area $= 2+2\sqrt{2}$ approximately $4.83$. Exterior angle $= 360/8 = 45^\circ$. The $8$ vertices lie on a circle.
Key Insight
A regular octagon inscribed in a square: if the square has side $a$, and equal right triangles of leg $x$ are cut from each corner, then $s = a - 2x$ and the triangle legs satisfy $x = s/\sqrt{2}$. Setting $s = a - 2(s/\sqrt{2})$ gives $a = s(1 + \sqrt{2})$, so $s = a/(1+\sqrt{2}) = a(\sqrt{2}-1)$.
Definition
A regular octagon has vertices at $\{e^{i\pi k/4} : k=0,\ldots,7\}$ ($8$th roots of unity). Its symmetry group is $D_8$ (dihedral, order $16$). Area $= 2(1+\sqrt{2})s^2$. The regular octagon is NOT a Platonic solid face (Platonic solids use triangles, squares, pentagons only), but it appears in the truncated cube and truncated octahedron (Archimedean solids).
Example
The $8$th roots of unity are $\{1, e^{i\pi/4}, i, e^{3i\pi/4}, -1, e^{5i\pi/4}, -i, e^{7i\pi/4}\}$, which simplify to $\{1, (1+i)/\sqrt{2}, i, (-1+i)/\sqrt{2}, -1, (-1-i)/\sqrt{2}, -i, (1-i)/\sqrt{2}\}$.
Key Insight
The regular octagon's interior angle of $135^\circ$ means three octagons at a vertex give $3\times135 = 405^\circ > 360$, so the regular octagon cannot tile the plane alone. However, octagons and squares together tile the plane (the "truncated square tiling"), appearing in Islamic geometric art and modern architecture.