1. What a polygon is
A closed plane figure bounded by three or more straight line segments — the sides — joined end-to-end so that each endpoint (a vertex) is shared by exactly two sides, and the sides do not cross.
Every part of that definition is doing work. Closed rules out a zig-zag path that doesn't loop back. Straight segments rules out circles and arcs. Three or more sets the floor — two segments can't enclose any area. And non-crossing distinguishes a simple polygon from a self-intersecting one like a pentagram (which is sometimes called a polygon, sometimes not, depending on who's writing).
Convex vs concave
A polygon is convex if every interior angle is less than $180°$ — equivalently, any line segment drawn between two points inside the polygon stays entirely inside. A concave (or non-convex) polygon has at least one interior angle greater than $180°$, which produces a "dent" you can see at a glance.
Stretch a rubber band tight around the polygon's vertices. If the band touches every vertex, the polygon is convex. If at least one vertex sits inside the rubber band, it's concave — that vertex is exactly where the dent is.
2. Naming polygons
Polygons are named after their number of sides, mostly from Greek roots. The first ten have everyday names; past that, mathematicians usually just say "n-gon."
| Sides | Name | Sides | Name |
|---|---|---|---|
| $3$ | Triangle | $8$ | Octagon |
| $4$ | Quadrilateral | $9$ | Nonagon (or enneagon) |
| $5$ | Pentagon | $10$ | Decagon |
| $6$ | Hexagon | $12$ | Dodecagon |
| $7$ | Heptagon | $n$ | n-gon |
"Quadrilateral" is the odd one out — Latin (quadri- + latus, "four sides") rather than Greek. "Tetragon" exists but is rarely used. For everything from pentagon onward the Greek prefix wins.
The name says nothing about the shape's regularity or convexity. A "hexagon" can be a perfect honeycomb cell or an irregular six-sided blob — both qualify.
3. Regular vs irregular
A polygon that is both equilateral (all sides equal in length) and equiangular (all interior angles equal). Either condition alone is not enough.
For triangles the two conditions are equivalent — equilateral and equiangular are the same triangle. Past three sides they diverge, and the trap is real:
- A rhombus has four equal sides but its angles can be anything from a near-degenerate squash to a square. Equilateral, not equiangular.
- A rectangle has four equal angles (all $90°$) but its sides come in two different lengths. Equiangular, not equilateral.
- Only the square satisfies both, which is why it's the unique regular quadrilateral.
So when you see "regular pentagon," picture a stop-sign-style shape where every side and every angle is identical — not just "a five-sided thing."
4. Sum of interior angles
The interior angles of any simple polygon — regular or not, convex or concave (as long as you measure reflex angles consistently) — sum to a value that depends only on the number of sides:
$$ S(n) = (n - 2) \cdot 180° $$So a triangle's angles sum to $180°$, a quadrilateral's to $360°$, a pentagon's to $540°$, and so on. Each extra side adds another $180°$.
Why? Triangulate from a vertex.
Pick any one vertex of an $n$-sided convex polygon and draw every possible diagonal from it. You get $n - 2$ non-overlapping triangles that tile the polygon. Each triangle contributes $180°$ to the total interior-angle sum, and those angles together add up to exactly the polygon's interior angles. Hence $S(n) = (n - 2) \cdot 180°$.
For a regular $n$-gon, each interior angle is just the total divided by $n$:
$$ \text{interior angle of a regular } n\text{-gon} \;=\; \frac{(n - 2) \cdot 180°}{n} $$So a regular hexagon's angles are each $\tfrac{4 \cdot 180°}{6} = 120°$, and a regular octagon's are each $\tfrac{6 \cdot 180°}{8} = 135°$.
Exterior angles always sum to $360°$
An exterior angle at a vertex is the supplement of the interior angle — the angle you'd turn through if you walked along the boundary and turned at that corner. For any simple convex polygon, one exterior angle per vertex, the sum is always:
$$ \sum \text{exterior angles} = 360° $$Intuitively, walking once around any convex polygon you make one complete turn — $360°$ — regardless of how many sides you took. For a regular $n$-gon each exterior angle is $360°/n$, so a regular hexagon's exterior angles are $60°$ apiece and a regular octagon's are $45°$.
Diagonals
A diagonal is a line segment joining two non-adjacent vertices. From any one vertex of an $n$-gon you can draw $n - 3$ diagonals (you can't connect to yourself or to your two neighbours). Counting every pair of non-adjacent vertices once gives the total:
$$ \text{diagonals of an } n\text{-gon} \;=\; \frac{n(n - 3)}{2} $$A quadrilateral has $\tfrac{4 \cdot 1}{2} = 2$ diagonals, a pentagon $\tfrac{5 \cdot 2}{2} = 5$, a hexagon $\tfrac{6 \cdot 3}{2} = 9$, and so on. (Triangles have zero — every pair of vertices is already connected by a side.)
5. Quadrilaterals: the family
Quadrilaterals are where the naming game gets dense. Six names cover almost every quadrilateral you'll meet, and each one says something specific about which sides, angles, or pairs are equal.
Definitions
A quadrilateral with both pairs of opposite sides parallel. Opposite sides are automatically equal in length, and opposite angles are equal.
A parallelogram with four right angles. Sides come in two equal pairs (length and width), and diagonals are equal in length.
A parallelogram with four equal sides. Diagonals are perpendicular and bisect each other (and the angles at the vertices).
A quadrilateral that is both a rectangle and a rhombus — four equal sides and four right angles. The unique regular quadrilateral.
A quadrilateral with at least one pair of parallel sides. (Under the "exactly one" definition, parallelograms are excluded; under the "at least one" definition, they're included — see the pitfalls section.)
A quadrilateral with two pairs of adjacent equal sides (rather than opposite). Its diagonals are perpendicular, and one diagonal bisects the other.
6. The quadrilateral hierarchy
Most of these shapes aren't disjoint categories — they nest. Every square is a rectangle (it satisfies the rectangle definition), and also a rhombus, and also a parallelogram, and (under the inclusive definition) a trapezoid. Each step up the chain adds a constraint.
Read it as nested boxes: every shape that fits in an inner box automatically fits in every outer box. A square is a rectangle, a rhombus, a parallelogram, a trapezoid (inclusive), and a quadrilateral — all at once. The reverse direction doesn't hold: most rectangles aren't squares.
| Shape | Defining extra property over a generic quadrilateral |
|---|---|
| Trapezoid (inclusive) | At least one pair of parallel sides |
| Parallelogram | Both pairs of opposite sides parallel |
| Rectangle | Parallelogram with four right angles |
| Rhombus | Parallelogram with four equal sides |
| Square | Both a rectangle and a rhombus |
| Kite | Two pairs of adjacent equal sides (not a parallelogram in general) |
If you prove a theorem about parallelograms — say, "opposite sides are equal" — it automatically applies to rectangles, rhombi, and squares. You don't have to re-prove it three more times. The hierarchy is a compression of redundant work.
7. Common pitfalls
The exclusive definition says a trapezoid has exactly one pair of parallel sides, which excludes parallelograms. The inclusive definition says at least one pair, which includes parallelograms, rectangles, and squares. US K-12 textbooks lean exclusive; most university mathematicians lean inclusive. Always check which one the source you're reading uses.
"Is that shape a rectangle or a square?" is a false choice. A square satisfies every property of a rectangle (four right angles, opposite sides equal) and every property of a rhombus (four equal sides). It's both — and you'll lose marks on a classification problem if you call a square "not a rectangle."
Regular is a strong claim. A regular pentagon has all five sides equal and all five interior angles equal to $108°$. A house-shaped pentagon (rectangle with a triangular roof) has five sides and isn't remotely regular. Don't use the word "regular" unless you really mean it.
The formula $S(n) = (n - 2) \cdot 180°$ is for simple polygons — sides that don't cross. A pentagram (five-pointed star) has five vertices, but its sides cross, and its interior angles sum to $180°$, not $540°$. If a polygon self-intersects, the formula no longer applies.
8. Worked examples
Try each one before opening the solution. The point is to internalize the formula and the hierarchy, not to memorize answers.
Example 1 · Sum of interior angles of an octagon
Step 1. Apply the formula with $n = 8$:
$$ S(8) = (8 - 2) \cdot 180° = 6 \cdot 180° = 1080° $$So the interior angles of any octagon — regular or not — sum to $1080°$.
Example 2 · Each interior angle of a regular decagon
Step 1. Total interior-angle sum with $n = 10$:
$$ S(10) = (10 - 2) \cdot 180° = 1440° $$Step 2. Divide by $n = 10$ because all angles are equal:
$$ \frac{1440°}{10} = 144° $$Each interior angle of a regular decagon is $144°$.
Example 3 · How many sides does a polygon have if its interior angles sum to $1800°$?
Step 1. Set up the equation:
$$ (n - 2) \cdot 180° = 1800° $$Step 2. Divide both sides by $180°$:
$$ n - 2 = 10 \quad\Longrightarrow\quad n = 12 $$So it's a dodecagon (12 sides).
Example 4 · Classify: four equal sides, no right angles
"Four equal sides" matches the definition of a rhombus. Since the angles aren't right, it isn't a square. But it is a parallelogram (a rhombus is a parallelogram with all sides equal), and under the inclusive definition it's also a trapezoid.
Most precise single label: rhombus (non-square).
Example 5 · A quadrilateral has interior angles $80°$, $100°$, and $90°$. What is the fourth?
Step 1. Sum for $n = 4$:
$$ S(4) = (4 - 2) \cdot 180° = 360° $$Step 2. Subtract the known angles:
$$ 360° - 80° - 100° - 90° = 90° $$The fourth angle is $90°$. Two right angles and a pair of complementary-ish angles — this could be a right trapezoid, for instance, but the angles alone don't pin down the exact shape.