Mathwords logoMathwords

Polygon

Polygon

A closed plane figure for which all sides are line segments. The name of a polygon describes the number of sides. A polygon which has all sides mutually congruent and all angles mutually congruent is called a regular polygon.

 

Number of sides Polygon name

An irregular pentagon with five unequal sides and angles, wider at the top and narrowing toward the bottom right.
pentagon

A regular undecagon (11-sided polygon) with equal sides and equal angles.
regular undecagon

Irregular octagon with eight sides, shown as an uneven polygon with indentations, representing a non-regular octagon example.
octagon

3 triangle
4 quadrilateral
5 pentagon
6 hexagon
7 heptagon
8 octagon
9 nonagon
10 decagon
11 undecagon
12 dodecagon
n n-gon
   

 

See also

Area of a regular polygon, area of a convex polygon

Key Formula

S=(n−2)×180°S = (n - 2) \times 180°
Where:
  • SS = Sum of interior angles of the polygon
  • nn = Number of sides (and also the number of angles)

Worked Example

Problem: Find the sum of the interior angles of a hexagon, and determine the measure of each interior angle if the hexagon is regular.
Step 1: Identify the number of sides. A hexagon has 6 sides, so n = 6.
n=6n = 6
Step 2: Apply the interior angle sum formula.
S=(6−2)×180°=4×180°=720°S = (6 - 2) \times 180° = 4 \times 180° = 720°
Step 3: For a regular hexagon, all six angles are equal. Divide the total by the number of angles.
Each angle=720°6=120°\text{Each angle} = \frac{720°}{6} = 120°
Answer: The sum of interior angles of a hexagon is 720°. If the hexagon is regular, each interior angle measures 120°.

Another Example

Problem: A regular polygon has interior angles that each measure 140°. How many sides does it have?
Step 1: In a regular polygon, each interior angle equals the total sum divided by the number of sides. Set up the equation.
(n−2)×180°n=140°\frac{(n - 2) \times 180°}{n} = 140°
Step 2: Multiply both sides by n to clear the fraction.
(n−2)×180=140n(n - 2) \times 180 = 140n
Step 3: Distribute and solve for n.
180n−360=140n180n - 360 = 140n
Step 4: Subtract 140n from both sides and solve.
40n=360  ⟹  n=940n = 360 \implies n = 9
Answer: The polygon has 9 sides, so it is a nonagon.

Frequently Asked Questions

Is a circle a polygon?
No. A polygon must be made entirely of straight line segments. A circle is a curved figure with no straight sides, so it does not meet the definition of a polygon.
What is the difference between a regular polygon and an irregular polygon?
A regular polygon has all sides the same length and all interior angles the same measure. An irregular polygon has sides or angles (or both) that differ in size. For instance, a square is a regular quadrilateral, while a rectangle with unequal side lengths is an irregular quadrilateral.

Convex polygon vs. Concave polygon

A convex polygon has all interior angles less than 180°, meaning no part of the shape "dents" inward. Every line segment connecting two interior points stays inside the figure. A concave polygon has at least one interior angle greater than 180° (a reflex angle), creating an inward dent. Both are valid polygons, but many standard formulas (like the simple interior angle formula for regular polygons) assume convex shapes.

Why It Matters

Polygons are the foundation of geometry. Nearly every flat shape you encounter—floor tiles, road signs, building footprints—is a polygon or can be approximated by one. Understanding polygons also leads directly to computing area, perimeter, and angle measures, which are essential skills in architecture, engineering, computer graphics, and everyday problem-solving.

Common Mistakes

Mistake: Thinking that any closed shape is a polygon.
Correction: A polygon must have only straight sides. Shapes with curves, like circles or ovals, are not polygons.
Mistake: Confusing the number of sides with the number of diagonals.
Correction: An n-sided polygon has n sides but n(n − 3)/2 diagonals. For example, a hexagon has 6 sides but 9 diagonals. These are different counts.

Related Terms