Pentagon
GeometryA 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. The interior angles of a pentagon add up to $540^\circ$.
Example
The US Department of Defense's headquarters building in Washington DC is shaped like a pentagon. A regular pentagon has all five sides equal and each angle measuring $108^\circ$.
Key Insight
The prefix "penta-" means five in Greek. A regular pentagon is special because its diagonals create a smaller regular pentagon inside, and the ratio of diagonal to side is the golden ratio $\phi = (1+\sqrt{5})/2$, approximately $1.618$.
Definition
A pentagon has $5$ sides, $5$ vertices, interior angle sum $= (5-2) \times 180 = 540^\circ$. A regular pentagon has interior angles of $108^\circ$ each and $5$ diagonals. The diagonals of a regular pentagon form a pentagram (five-pointed star), with the ratio of diagonal to side equal to the golden ratio.
Example
Regular pentagon with side $1$: diagonal $= \phi = (1+\sqrt{5})/2$ approximately $1.618$. Interior angle $= 108^\circ$, exterior angle $= 72^\circ$. Number of diagonals $= 5(5-3)/2 = 5$.
Key Insight
The golden ratio appears naturally in the regular pentagon: each diagonal is $\phi$ times the side length. Because of this, pentagons and pentagrams appear throughout art and architecture associated with aesthetic proportion. The Fibonacci sequence also converges to the golden ratio.
Definition
A regular pentagon has vertices at $\{e^{2\pi i k/5} : k=0,\ldots,4\}$ on the unit circle. It is constructible with compass and straightedge (since $5$ is a Fermat prime). Its symmetry group is $D_5$ (dihedral, order $10$). The diagonal-to-side ratio $\phi = (1+\sqrt{5})/2$ satisfies $\phi^2 = \phi + 1$, making it the positive root of $x^2 - x - 1 = 0$.
Example
The continued fraction of $\phi = 1 + 1/(1 + 1/(1 + \ldots))$ is the simplest infinite continued fraction, making $\phi$ the "most irrational" number - the hardest to approximate by rationals. This extremal property relates to the pentagon's role in the densest irrational spiral phyllotaxis patterns in nature.
Key Insight
The golden ratio $\phi$ embedded in the regular pentagon connects to an astonishing range of mathematics: Fibonacci sequences, continued fractions, optimal irrational approximation, quasicrystals (Penrose tilings use pentagons), and the icosahedron. The pentagon is a geometric seed of remarkable mathematical depth.