Irregular Polygon — Definition, Formula & Examples
An irregular polygon is a polygon that does not have all sides equal in length and all angles equal in measure. Most polygons you encounter in everyday life—like rectangles, scalene triangles, and random quadrilaterals—are irregular.
A polygon is classified as irregular if it fails to be equilateral (all sides congruent) or equiangular (all interior angles congruent), or both. Equivalently, a polygon that is not regular is irregular.
How It Works
To determine whether a polygon is irregular, check its side lengths and angle measures. If even one side differs in length from another, or one angle differs from another, the polygon is irregular. A rectangle, for instance, has all angles equal to but its sides are not all equal, so it is irregular. A rhombus has all sides equal but its angles are not all equal (unless it is a square), so it is also irregular.
Example
Problem: A quadrilateral has side lengths 3 cm, 5 cm, 3 cm, and 7 cm. Is it a regular or irregular polygon?
Step 1: Check whether all four sides are equal.
Step 2: Since the sides are not all the same length, the polygon cannot be regular.
Answer: The quadrilateral is an irregular polygon because its sides are not all congruent.
Why It Matters
Nearly every polygon in architecture, map boundaries, and design is irregular. Recognizing irregular polygons matters because you cannot use the shortcut formulas for regular polygons (like the apothem-based area formula) and must instead break the shape into simpler pieces such as triangles or rectangles to find its area.
Common Mistakes
Mistake: Thinking a polygon must have unequal sides AND unequal angles to be irregular.
Correction: A polygon is irregular if either its sides or its angles (or both) are not all equal. A rectangle has equal angles but unequal sides, so it is still irregular.