Irregular Polygons: Definition, Types & Area

TL;DR

Irregular polygons are closed flat shapes whose sides are not all equal and whose angles are not all equal — the opposite of regular polygons. This article defines them, lists the common types, shows that the interior-angle sum is still (n−2)×180°, and walks through finding the area by decomposition — splitting the shape into triangles and rectangles.

Last updated on June 15, 2022, 9 min read

What Is an Irregular Polygon?

An irregular polygon is a closed two-dimensional shape made of three or more straight sides in which the sides are not all equal in length and the interior angles are not all equal in measure. Even if just one side or one angle differs from the rest, the polygon is irregular.

Contrast that with a regular polygon, where every side has the same length and every interior angle has the same measure (an equilateral triangle, a square, a regular hexagon). A regular polygon needs both conditions; an irregular polygon is anything that fails either.

A reader question worth answering up front — is a rectangle an irregular polygon? Yes, usually. A (non-square) rectangle has all 90° angles but its sides are not all equal — length differs from width — so it fails the equal-sides condition and counts as irregular. Equal angles alone are not enough; a regular polygon needs equal sides too.

Irregular polygons can be convex (no interior angle exceeds 180°, no "dents") or concave (at least one interior angle is reflex, giving an inward dent). Both are irregular as long as the sides and angles are not all equal.

Types of Irregular Polygons

Because "irregular" just means "not all equal," most familiar shapes are irregular. The common named ones:

Scalene triangle — all three sides different lengths, all three angles different.
Right triangle — one 90° angle; its three sides are generally unequal.
Isosceles triangle — two equal sides, but the third differs, so the angles are not all equal.
Rectangle (non-square) — equal angles (90°) but unequal sides.
Irregular quadrilateral — a four-sided shape such as a general trapezoid or kite, with sides and angles that differ.
Irregular pentagon, hexagon, and beyond — any five- or six-sided (or more) shape whose sides and angles are not all equal.

The Angle Rules That Still Apply

Being irregular does not free a polygon from the angle rules — it just removes the "all equal" shortcut.

The sum of the interior angles of any n-sided polygon, regular or irregular, is:

S=(n−2)×180°.

This holds because the sum depends only on the number of sides, not on whether the shape is symmetric.

The sum of the exterior angles is also still 360° for an irregular polygon — exactly as for a regular one.

How to Find the Area of an Irregular Polygon

The reliable method is decomposition: break the shape into smaller pieces whose areas you can compute — usually triangles and rectangles — then add them up.

The method, step by step:

Divide the irregular polygon into non-overlapping triangles and rectangles by drawing in extra lines.
Find the area of each piece using its own formula — rectangle area is length × width, triangle area is 1/2 × base × height.
Add the areas of all the pieces. The total is the area of the irregular polygon.

Examples of Irregular Polygons

With the definition, the angle rules, and the decomposition method in place, here are some examples:

Example 1

Is a shape with sides 555 cm, 555 cm, 555 cm, and 777 cm a regular or irregular polygon? Three sides are equal but the fourth (777 cm) is not, so the sides are not all equal. The shape is an irregular polygon.

Example 2

Find each interior angle of an irregular quadrilateral whose interior angles are in the ratio 1:2:3:4. The four angles still sum to (4−2)×180°=360°. Split 360° into 1+2+3+4=10 parts: each part is 36°. So the angles are 36°, 72°, 108°, 144°.

Example 3

Find the missing interior angle of an irregular pentagon whose other four angles are 100°, 120°, 90°, and 130°. The five interior angles sum to (5−2)×180°=540°. Add the four known: 100° + 120° + 90° + 130° = 440°. The fifth is 540°−440°=100°.

Example 4

Find the area of an L-shaped polygon that is an 888 m × 6 m rectangle with a 333 m × 2 m rectangular notch cut out of one corner. Area = (8×6) − (3×2) = 48−6=42 m².

Example 5

An irregular polygon is split into a rectangle of area 24 cm² and two triangles of areas 9 cm² and 6 cm². Find its total area. Total = 24 + 9 + 6 = 39 cm².

Example 6

A composite garden is a rectangle 10 m × 4 m with a triangular flower bed of base 4 m and height 3 m attached to one short end. Find the total area. Rectangle area: 10×4=40 m². Triangle area: 1/2 × 4 × 3 = 6 m². Total: 40 + 6 = 46 m².

Why Irregular Polygons Matter Beyond the Classroom

Irregular polygons are the geometry of anything not mass-produced — which is most of the physical world:

Land surveying and real estate. Plots of land are almost never neat rectangles; their area is found by splitting the boundary into triangles.
Architecture and floor plans. A house rarely has a single rectangular footprint; architects compute floor area by breaking an irregular plan into rectangles.
Geography and mapping. The area of a state, a lake, or a forest is an irregular-polygon problem, handled today by the shoelace formula running on GPS-traced vertex coordinates.
Computer graphics. Any 3D model is built from irregular polygon meshes; rendering and physics engines compute lighting and collisions.

For a Grade 8 student, irregular polygons are where geometry becomes a tool for measuring whatever the real world hands you — by breaking the unfamiliar into the familiar.

Where Students Trip Up on Irregular Polygons

Mistake 1: Dividing a total by n to find one angle

Correct way: Adding known angles and subtracting from the total.

Mistake 2: Forgetting the angle-sum rule still applies

Correct way: Remember that the interior-angle sum is still (n−2)×180°.

Mistake 3: Overlapping pieces when decomposing for area

Correct way: Ensure pieces are non-overlapping and cover the whole shape.

Key Takeaways

Irregular polygons are shapes whose sides and angles are not all equal.
Many common real-world shapes are irregular.
The interior-angle sum is (n−2)×180° and the exterior-angle sum is 360°.
Find the area using decomposition.
Common mistakes include incorrectly dividing totals by n.