Pentagon
A pentagon is a polygon with five sides and five interior angles summing to 540 degrees.
Formula
\text{Interior angle sum} = 540^\circ; \text{each regular angle} = 108^\circ
Definition
A pentagon is a polygon with five sides and five corners, whose interior angles add up to $540^\circ$ ($=(5-2)\times180$); a regular pentagon has each angle measuring $108^\circ$, an exterior angle of $72^\circ$, and $5(5-3)/2=5$ diagonals, which form a pentagram (five-pointed star) whose diagonal-to-side ratio is the golden ratio $\phi=(1+\sqrt{5})/2$, satisfying $\phi^2=\phi+1$. Its vertices sit at $\{e^{2\pi ik/5}:k=0,\ldots,4\}$ on the unit circle, and it is constructible with compass and straightedge since $5$ is a Fermat prime, with symmetry group $D_5$ of order $10$.
Example
The US Department of Defense headquarters in Washington DC is shaped like a pentagon. For side $1$: diagonal $=\phi\approx1.618$, interior angle $108^\circ$, exterior angle $72^\circ$, and $5$ diagonals total. The continued fraction $\phi=1+1/(1+1/(1+\ldots))$ is the simplest infinite continued fraction, making $\phi$ the "most irrational" number, the hardest to approximate by rationals.
Key Insight
The prefix "penta-" means five in Greek, and a regular pentagon's diagonal-to-side ratio being the golden ratio is why pentagons and pentagrams appear throughout art and architecture associated with aesthetic proportion, alongside their connection to the Fibonacci sequence. This same golden ratio embedded in the pentagon connects to Fibonacci sequences, continued fractions, optimal irrational approximation, quasicrystals (Penrose tilings use pentagons), and the icosahedron, making the pentagon a geometric seed of remarkable mathematical depth.