Polygons — Mathematics Knowledge System

Overview

A polygon is a plane figure bounded by a finite chain of straight line segments (its sides) closing to form a loop, meeting only at their shared endpoints (the vertices). Triangles and quadrilaterals are the polygons with the fewest sides; the family continues with pentagons (5 sides), hexagons (6), heptagons (7), octagons (8), and generally an n-gon for n sides. A polygon is convex if every interior angle is less than 180° (no vertex 'caves in'); it is regular if all sides and all interior angles are equal. The interior angle sum of any simple polygon grows linearly with the number of sides, which is the single most important structural fact about polygons.

Intuition

Any simple polygon with n sides can be cut into exactly (n − 2) triangles by drawing diagonals from one vertex to every other non-adjacent vertex — a quadrilateral (n=4) splits into 2 triangles, a pentagon (n=5) into 3, a hexagon (n=6) into 4, and so on. Since each triangle contributes 180° of interior angle, the whole polygon's interior angles sum to (n−2)×180°. This is the same diagonal-splitting idea used for quadrilaterals, just extended to more sides — the triangle really is the atomic unit that all polygon angle sums are built from.

Interactive Graph

Formal Definition

Definition

For a simple polygon with n ≥ 3 sides, and for a regular polygon with n sides of length s and apothem (center-to-side distance) a:

S = (n-2)\times 180^\circ

Interior angle sum of an n-gon

\theta_{\text{interior}} = \frac{(n-2)\times 180^\circ}{n}

Each interior angle of a regular n-gon

\theta_{\text{exterior}} = \frac{360^\circ}{n}

Each exterior angle of a regular n-gon

A_{\text{regular}} = \tfrac{1}{2}\, n\, s\, a

Area of a regular n-gon (half perimeter times apothem)

Notation

Notation	Meaning
$n$	Number of sides (equivalently, vertices) of the polygon
$a$	Apothem: the perpendicular distance from the center to a side of a regular polygon
$S$	Sum of interior angles

Properties

Interior angle sum

S = (n-2)\times 180^\circ

Condition: Holds for every simple polygon with n sides, convex or not.

Example: A hexagon (n=6): S = (6-2)×180° = 720°.

Exterior angle sum

\text{The exterior angles of any convex polygon, one per vertex, always sum to } 360^\circ.

Regular polygon

\text{All sides equal in length and all interior angles equal.}

Example: A regular hexagon has interior angles of 120° each.

Convex vs. concave

\text{Convex: every interior angle} < 180^\circ. \text{ Concave: at least one interior angle} > 180^\circ.

Applications

Polygon meshes (built from triangles or quads) are the standard way to represent 3D surfaces in computer graphics and CAD.

Worked Examples

Apply the interior angle sum formula with n = 5.
$S = (5-2)\times 180^\circ = 3\times 180^\circ = 540^\circ$

Answer: 540°

Practice Problems

Difficulty 2/10

Find the sum of the interior angles of a heptagon (7 sides).

Difficulty 3/10

Find each interior angle of a regular decagon (10 sides).

Difficulty 5/10

A regular polygon-shaped tabletop has each interior angle equal to 150°. How many sides does the tabletop have?

Common Mistakes

Common Mistake

Using (n-2)×180° for the sum of exterior angles too.

The exterior angle sum of any convex polygon is always 360°, regardless of the number of sides — only the interior angle sum depends on n.

Common Mistake

Assuming all polygons with equal side lengths must be regular.

Regularity requires BOTH equal side lengths AND equal interior angles; a rhombus, for instance, has four equal sides but is regular only if it's a square.

Quiz

Summary

A polygon with n sides has interior angles summing to (n-2)×180°.
Any convex polygon's exterior angles (one per vertex) always sum to 360°.
A regular polygon has all sides and all interior angles equal; each interior angle is (n-2)×180°/n.
Any simple polygon can be triangulated into (n-2) triangles from a single vertex — the source of the angle-sum formula.
Convex polygons have all interior angles under 180°; concave polygons have at least one reflex (>180°) angle.

References

WebsiteWikipedia — Polygon