# 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: 1. **Divide** the irregular polygon into non-overlapping triangles and rectangles by drawing in extra lines. 2. **Find the area of each piece** using its own formula — rectangle area is length × width, triangle area is 1/2 × base × height. 3. **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.