A polygon is a two-dimensional closed figure formed by a finite number of straight line segments joined end to end. Each segment is a side (or edge); each point where two sides meet is a vertex (corner). A figure qualifies as a polygon only if it is closed, lies flat in one plane, and has no curved or crossing sides. A triangle is the smallest polygon, with three sides; from there the count rises with no upper limit.
By the end you will be able to tell a polygon from a non-polygon at a glance, name a polygon from its side count, and compute the sum of its interior angles. A polygon's full classification lives in types of polygon; the special case where no corner caves inward is the convex polygon.
What Makes a Figure a Polygon (And What Disqualifies It)
A shape passes the polygon test only if it meets all four conditions at once.
It is closed. The sides form an unbroken loop with no open ends. An L-shaped path of two segments is not a polygon.
Every side is a straight line segment. A circle and a semicircle are out, because their boundaries curve.
It is flat (planar). All vertices lie in one plane. A cube is a solid, not a polygon, though each of its faces is one.
The sides do not cross. A figure whose sides intersect themselves (like a pentagram) is a complex or self-intersecting polygon; the everyday meaning of "polygon" is the simple polygon, whose sides meet only at shared vertices.
A common point of confusion: is a circle a polygon? No. A circle has one continuous curved boundary, so it has no straight sides and no vertices, and it fails the very first requirement.
Parts of a Polygon
Every polygon shares the same vocabulary, and naming these parts early makes the rest of geometry readable.
Side (edge): one of the straight segments forming the boundary.
Vertex: a corner where two sides meet. A polygon with $n$ sides has exactly $n$ vertices.
Interior angle: the angle inside the polygon at a vertex. These are the angles studied in interior angles.
Exterior angle: the angle between a side and the extension of the next side; see exterior angles of a polygon.
Diagonal: a segment joining two non-adjacent vertices.
How Polygons are Named and Classified
Polygons are sorted along three independent axes, and a single polygon carries a label on each.
By number of sides. The Greek prefix names the polygon. A 3-gon is a triangle, a 4-gon a quadrilateral, a 5-gon a pentagon, a 6-gon a hexagon, an 8-gon an octagon. For a side count with no common name, mathematicians simply write "$n$-gon."
Sides | Name | Interior-angle sum |
|---|---|---|
3 | Triangle | $180°$ |
4 | Quadrilateral | $360°$ |
5 | Pentagon | $540°$ |
6 | Hexagon | $720°$ |
8 | Octagon | $1080°$ |
10 | Decagon | $1440°$ |
$n$ | $n$-gon | $(n-2)\times 180°$ |
By regularity. A regular polygon has all sides equal and all angles equal (a square, an equilateral triangle). An irregular polygon breaks at least one of those.
By shape. A convex polygon has every interior angle less than $180°$, so no corner caves inward. A concave polygon has at least one reflex interior angle (greater than $180°$), so part of it dents inward.
The Interior-Angle Sum Formula, And Where it Comes From
The single most useful polygon fact is the sum of its interior angles. Rather than memorise it, build it.
Pick any vertex of a polygon and draw every diagonal from it. For an $n$-sided polygon, those diagonals cut the interior into exactly $(n-2)$ triangles. Each triangle's angles add to $180°$, and together they account for every interior angle of the polygon. So:
$$\text{Sum of interior angles} = (n-2)\times 180°$$
For a regular polygon, all $n$ angles are equal, so each one measures:
$$\text{Each interior angle} = \frac{(n-2)\times 180°}{n}$$
Here $n$ is the number of sides, $(n-2)$ is the number of triangles the diagonals create, and $180°$ is the angle sum of one triangle. The exterior angles tell an even simpler story: they always sum to $360°$, no matter how many sides the polygon has.
Examples of Polygons
Example 1
Name the polygon with 7 sides and find the sum of its interior angles.
A 7-sided polygon is a heptagon.
$$\text{Sum} = (7-2)\times 180° = 5 \times 180° = 900°$$
The seven interior angles together measure $900°$.
Example 2
A student says a regular pentagon's interior angle is $\dfrac{540°}{4} = 135°$. Find the error and the correct value.
A natural first move is to divide the angle sum by the number of triangles, $4$. Try it: $540° \div 4 = 135°$. But that splits the total across the triangles, not the angles — and a pentagon has five interior angles, not four.
Divide by the number of sides instead:
$$\text{Each angle} = \frac{540°}{5} = 108°$$
Each interior angle of a regular pentagon is $108°$, not $135°$. The $(n-2)$ counts triangles; the divisor for "each angle" is always $n$.
Example 3
Is a figure with four straight sides where two sides cross (a "bowtie") a simple polygon?
The figure has four straight segments, so it is built from the right pieces.
But two of its sides intersect away from a vertex, so it is self-intersecting.
It is therefore a complex polygon, not a simple one. The everyday meaning of "polygon" excludes it.
Example 4
A regular polygon has each interior angle equal to $150°$. How many sides does it have?
Set the regular-polygon angle formula equal to $150°$:
$$\frac{(n-2)\times 180°}{n} = 150°$$
Multiply both sides by $n$:
$$(n-2)\times 180° = 150°,n$$ $$180n - 360 = 150n$$ $$30n = 360$$ $$n = 12$$
The polygon is a regular 12-gon (a dodecagon).
Example 5
Find the number of diagonals in a hexagon.
The number of diagonals of an $n$-gon is $\dfrac{n(n-3)}{2}$.
$$\frac{6(6-3)}{2} = \frac{6 \times 3}{2} = \frac{18}{2} = 9$$
A hexagon has 9 diagonals.
Example 6
A floor tile is a regular octagon. The remaining gaps between four such tiles are small squares. Show why the octagons and squares fit together with no gaps.
A regular octagon's interior angle is $\dfrac{(8-2)\times 180°}{8} = 135°$.
At the corner where tiles meet, the angles must add to a full $360°$.
Two octagon corners and one square corner give $135° + 135° + 90° = 360°$.
The angles close exactly, so the octagon-and-square pattern tiles a floor with no overlap and no gap. This is why the octagon-square tiling is one of the classic floor patterns.
Where Polygons Earn Their Keep: From Honeycombs to Game Graphics
Polygons are not just textbook shapes — they are the building blocks the physical and digital worlds are assembled from, because straight sides are cheap to make and easy to compute.
Tiling and architecture. Only certain regular polygons tile a flat plane edge to edge with no gaps: the triangle, the square, and the hexagon. Bees build hexagonal honeycombs because the hexagon stores the most area for the least wax — a result that holds purely because of the polygon's angles.
Computer graphics. Every 3-D model in a video game or film is a mesh of polygons, almost always triangles, because a triangle is always flat and always convex, which makes lighting and rendering fast and unambiguous.
Why the straight-side rule matters. The destination here is computation. Curves are expensive to describe; a polygon is fully captured by a short list of vertex coordinates. That is why engineers approximate even smooth shapes with many-sided polygons.
The hexagonal efficiency of the honeycomb was proved rigorously only in 1999, when Thomas Hales settled the honeycomb conjecture — a problem the bees had "solved" for millions of years.
Polygons Mistakes To Watch For
Mistake 1: Counting a curved or open figure as a polygon
Where it slips in: When a figure looks shape-like, students label it a polygon without checking the closed-and-straight conditions.
Don't do this: Call a semicircle, an arc, or an open zig-zag a polygon because "it has straight parts."
The correct way: Run all four checks — closed, straight sides, flat, non-crossing. A semicircle fails on the curve; an open zig-zag fails on closure. The student who eyeballs shapes instead of checking the definition will reliably mislabel the first curved figure on a test.
Mistake 2: Dividing the angle sum by $(n-2)$ instead of $n$
Where it slips in: Finding each interior angle of a regular polygon, right after computing the sum.
Don't do this: Write $\dfrac{(n-2)\times 180°}{n-2} = 180°$, which absurdly makes every angle a straight angle.
The correct way: The sum is $(n-2)\times 180°$, but it is shared among $n$ angles, so divide by $n$. The student who memorised the formula as one blur, rather than understanding that $(n-2)$ counts triangles, makes this slip the moment the two numbers sit side by side.
Mistake 3: Assuming every polygon is convex
Where it slips in: Applying convex-only reasoning (every diagonal stays inside) to a concave polygon.
The correct way: Check for a reflex angle first. A polygon with even one interior angle above $180°$ is concave, and some of its diagonals leave the figure. The convex case is the one where every diagonal stays inside.
Key Takeaways
A polygon is a closed, flat figure made of three or more straight, non-crossing sides.
Polygons carry three labels at once: a side-count name, regular or irregular, and convex or concave.
The interior angles sum to $(n-2)\times 180°$ because diagonals from one vertex split the polygon into $(n-2)$ triangles.
Each interior angle of a regular polygon is $\dfrac{(n-2)\times 180°}{n}$, and exterior angles always sum to $360°$.
Circles and open or self-crossing figures are not simple polygons.
A Practical Next Step
Practice these problems to solidify your understanding. For each shape, name it by side count, decide whether it is regular or irregular and convex or concave, then compute its interior-angle sum.
A polygon has 9 sides. Find the sum of its interior angles. (Answer to Question 1: $(9-2)\times 180° = 1260°$.)
A regular polygon has each interior angle equal to $144°$. How many sides does it have? (Answer to Question 2: $n = 10$, a decagon.)
To take polygons further with a teacher, explore Bhanzu's geometry tutor, our high school math tutor sessions, or math classes online. To watch a trainer build the interior-angle formula live, you can book a free demo class.
Read More
Geometric shapes — the wider family of 2-D and 3-D shapes polygons belong to.
Quadrilaterals — the four-sided polygons and how they are classified.
Types of triangles — the smallest polygon, sorted by sides and angles.
Square in geometry — a regular, convex, four-sided polygon worked in detail.
Angles in a pentagon — the interior-angle rule worked out for the five-sided case.
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