Octagon
An 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, whose angles add up to $1080^\circ$ ($=(8-2)\times180$); a regular octagon has each angle $135^\circ$, $20$ diagonals, and can be formed by cutting equal isosceles right triangles from the corners of a square. Its vertices sit at $\{e^{i\pi k/4}:k=0,\ldots,7\}$ ($8$th roots of unity), symmetry group $D_8$ of order $16$, area $=2(1+\sqrt{2})s^2$; it is not a Platonic solid face (those use only triangles, squares, and pentagons) but appears in the truncated cube and truncated octahedron.
Example
A stop sign is a regular octagon; cutting the corners off a square produces an octagon-like shape. For side $s=1$: area $=2+2\sqrt{2}\approx4.83$, exterior angle $=45^\circ$. If a square of side $a$ has equal right-triangle legs $x$ cut from each corner, then $s=a-2x$ with $x=s/\sqrt{2}$, giving $a=s(1+\sqrt{2})$, so $s=a/(1+\sqrt{2})=a(\sqrt{2}-1)$.
Key Insight
The prefix "octa-" means eight in Greek (as in octopus), and a regular octagon looks like a circle with its corners cut flat, each interior angle $135^\circ$ sitting halfway between a square's $90^\circ$ and a flat line's $180^\circ$. Since $3\times135=405^\circ$ exceeds $360^\circ$, a regular octagon cannot tile the plane alone, but octagons and squares together do (the "truncated square tiling"), a pattern that appears in Islamic geometric art and modern architecture.