Polygons: Definition, Types, Properties, and Formulas

TL;DR

A polygon is a closed, flat shape made of three or more straight sides joined end to end, with no curves and no gaps. This article defines the polygon, classifies it by sides and by shape, derives the interior-angle sum formula (n−2)×180° and works through examples and the mistakes students make most.

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 nnn sides has exactly nnn 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 "nnn-gon."

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 memorize 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:

Sum of interior angles=(n−2)×180°

For a regular polygon, all n angles are equal, so each one measures:

Each interior angle=(n−2)×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.
Sum=(7−2)×180°=5×180°=900°.

Example 2

A student says a regular pentagon's interior angle is 540°/4=135°. Find the error and the correct value.
A natural first move is to divide the angle sum by the number of angles, 5. Divide by the number of sides instead:
Each angle=540°/5=108°.
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.

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°:
(n−2)×180°/n=150°.
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 n(n−3)/2.
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 (8−2)×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°.

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. 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.

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.
The correct way: Run all four checks — closed, straight sides, flat, non-crossing.

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.
The correct way: The sum is (n−2)×180°, but it is shared among n angles, so divide by n.

Mistake 3: Assuming every polygon is convex

The correct way: Check for a reflex angle first. A polygon with even one interior angle above 180° is concave.

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)×180° because diagonals from one vertex split the polygon into (n−2) triangles.
• Each interior angle of a regular polygon is (n−2)×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.

1. A polygon has 9 sides. Find the sum of its interior angles. (Answer: 1260°)

2. A regular polygon has each interior angle equal to 144°. How many sides does it have? (Answer: 10, a decagon.)