Polygon – Shape, Types, Formulas, Examples
reviewed by Jo-ann Caballes
Updated on October 1, 2026
A polygon is a closed, two-dimensional shape made up of three or more straight sides connected end to end, with no curves or gaps — a triangle, a square, and a stop sign are all polygons. The word comes from the Greek poly (“many”) and gon (“angle”), since every polygon has at least three angles. This guide covers polygon types, the angle formulas that describe them, and worked examples — solid groundwork for students working with elementary math tutors on geometry and shapes.
Still finding polygons tricky? Our 1:1 geometry tutors explain it step by step in live 1:1 lessons.
What Is a Polygon?
A polygon is any plane (flat), closed shape made of straight, connected line segments, with at least three sides and three angles. A shape with even one curved edge or one open gap in its outline is not a polygon.
A polygon looks like a notebook, a door, or a stop sign — all shapes bounded entirely by straight sides. A circle is not a polygon, since it’s made of a curve rather than straight sides, and neither is a shape like a slice of pizza, since its curved crust edge breaks the straight-sides rule.
A polygon can be identified by these characteristics:
- It has at least 3 sides
- It is two-dimensional — it has only length and width
- It has no open spaces or curved edges
- It is made up entirely of straight lines and sides
Expert Math Tutors for Every Level
Connect with professional and certified educators who provide specialized support for every grade level, from elementary school basics to advanced algebra
Quick Knowledge Check
Test what you’ve just learned with this short quiz before moving on.
Types of Polygons by Number of Sides
A polygon can have 3, 4, 5, or more sides, and each side count has its own name:
| Number of Sides | Name of Polygon |
| 3 | Trigon or Triangle |
| 4 | Tetragon or Quadrilateral |
| 5 | Pentagon |
| 6 | Hexagon |
| 7 | Heptagon |
| 8 | Octagon |
| 9 | Nonagon |
| 10 | Decagon |
Fun fact: most of these prefixes are Greek in origin — “tri” means three, “tetra” means four, “penta” means five, and so on, the same roots behind many math terms.
Also, here are some examples of shapes that do not qualify as polygons:
Help your child reach their full potential!
Answer a few quick questions about your child’s learning, and we’ll recommend next steps.
Regular and Irregular Polygons
A regular polygon has all sides and all angles equal — a square is a common example.
An irregular polygon has sides and angles that are not all equal, like an obtuse triangle with three different side lengths.
Convex and Concave Polygons
Polygons can also be classified by their interior angles: convex or concave.
A convex polygon has every interior angle less than 180°, so none of its sides bend inward. Every regular polygon is convex, but plenty of convex polygons are irregular too — a general rectangle or an uneven convex pentagon are both convex without being regular.
A concave polygon has at least one interior angle greater than 180°, creating an inward dent in the shape. Because a regular polygon is always convex, a concave polygon is always irregular.
There are other, more advanced polygon categories too — simple, complex, equiangular, tangential, and more — but convex, concave, regular, and irregular cover what most students need at this level.
Interior and Exterior Angles of a Polygon
Every polygon has two kinds of angles: interior and exterior. Interior angles are found inside the polygon, where two sides meet. Exterior angles are formed outside the polygon, between one side and the extension of the next.
Sum of the Interior Angles of a Polygon
To find the sum of a polygon’s interior angles, subtract 2 from the number of sides, then multiply by 180°:
(n − 2) × 180°
For a polygon with 5 sides (a pentagon):
- (5 − 2) × 180° = 3 × 180° = 540°
So a pentagon’s interior angles always add up to 540°.
Sum of the Exterior Angles of a Polygon
The exterior angles of any polygon — regular or irregular — always add up to 360°. For a regular polygon, since every exterior angle is equal, you can find a single exterior angle by dividing 360° by the number of sides. For an irregular polygon, each exterior angle is different, so finding one requires knowing the total number of sides and the value of the other exterior angles. This connects closely to the perimeter of a polygon, since both describe how a shape’s sides and angles fit together.
Solved Examples
Example 1: Which of these is a tetragon: a shape with 3 sides, a shape with 4 sides, or a shape with 6 sides? Since “tetra” means four, the shape with 4 sides is the tetragon (also called a quadrilateral).
Example 2: Name each polygon based on its sides or properties: a polygon with 7 sides, a polygon with 9 sides, a polygon with equal sides and angles, a polygon with 3 sides, a polygon with 5 sides, and a polygon with 10 sides.
- 7 sides → Heptagon
- 9 sides → Nonagon
- Equal sides and angles → Regular polygon
- 3 sides → Triangle
- 5 sides → Pentagon
- 10 sides → Decagon
Example 3: A polygon’s interior angles add up to 540°. How many sides does it have? Using 540° = (n − 2) × 180°: n − 2 = 540 ÷ 180 = 3, so n = 3 + 2 = 5. The polygon has 5 sides — it’s a pentagon.
Example 4: Find the sum of the interior angles of a polygon with 8 sides. (8 − 2) × 180° = 6 × 180° = 1,080°. An octagon’s interior angles add up to 1,080°.
Want to start with a first lesson? Meet an online math tutor from our team.
Polygon Worksheets
With Brighterly, young students can further their learning about shapes, polygons, and geometry through these worksheets:
- Identifying shapes worksheets
- Types of triangles worksheets
- Measuring angles worksheets
- 5th grade geometry worksheets
Frequently Asked Questions About Polygons
What Is a Polygon?
A polygon is a closed, two-dimensional shape made of three or more straight sides connected end to end, with no curves or gaps. Triangles, squares, and hexagons are all polygons, while circles and any shape with a curved edge are not.
What Is the Difference Between a Regular and an Irregular Polygon?
A regular polygon has all sides and all angles equal, like a square or an equilateral triangle. An irregular polygon has sides or angles that are not all equal, such as a rectangle (equal angles, but two different side lengths) or a scalene triangle.
Is Every Convex Polygon a Regular Polygon?
No. Convex only means every interior angle is less than 180°, with no sides bending inward — it says nothing about whether the sides or angles are equal. A general rectangle, for example, is convex but not regular, since its two pairs of sides have different lengths.
How Do You Find the Sum of a Polygon’s Interior Angles?
Subtract 2 from the number of sides, then multiply by 180°: (n − 2) × 180°. A hexagon (6 sides), for example, has interior angles that add up to (6 − 2) × 180° = 720°.
Do the Exterior Angles of Every Polygon Add Up to the Same Total?
Yes. The exterior angles of any polygon, regular or irregular, always sum to 360°. For a regular polygon, dividing 360° by the number of sides gives the size of each individual exterior angle.
What’s the Smallest Possible Polygon?
A triangle, with 3 sides, is the polygon with the fewest possible sides. A closed shape needs at least three straight sides to enclose an area, so two sides can never form a polygon.
Problems with Geometry?
- Does your child need extra assistance with grasping geometry?
- Start learning with an online tutor.
Choose kid's grade
Is your child finding it hard to understand geometry? An online tutor could be of help.
Book FREE lesson
