# Area Of Polygons - Definition, Formula, and Examples

## TL;DR

The area of a polygon is the flat space it encloses. A regular polygon uses A=12,P,a while an irregular polygon is found by splitting it into triangles or by the shoelace formula A=\frac{1}{2}\left\|\sum (x_i y_{i+1} - x_{i+1} y_i)\right\|. This article defines both routes and works through examples.

## Why There Is No Single "Area Of A Polygon" Formula

The **area of a polygon** is the amount of two-dimensional space enclosed by its sides. The method depends on the polygon's regularity: a **regular polygon** (all sides and all angles equal) uses the apothem formula A=\frac{1}{2},P,a, while an **irregular polygon** is handled either by **triangulation** (splitting it into triangles and adding their areas) or by the **shoelace formula** when the corner coordinates are known.

## Route 1 - Regular Polygons: The Apothem Formula

A **regular polygon** has all sides equal and all angles equal, which gives it a single centre equidistant from every side. That distance - the **apothem** - is the key.

A=\frac{1}{2},P,a

If you know only the side length s and the number of sides n, the apothem is a=\dfrac{s}{2\tan(180°/n)}. This single formula covers the equilateral triangle, square, pentagon, hexagon, and every regular n-gon, once the apothem is known.

## Route 2 - Irregular Polygons: Triangulation And The Shoelace Formula

An **irregular polygon** has no single centre distance, so the apothem formula fails. Two reliable methods handle it.

**Triangulation.** Pick one vertex and draw diagonals to every non-adjacent vertex. This cuts an n-sided polygon into n−2 triangles.

**Shoelace formula.** The area is:

A=\frac{1}{2}\left\|\sum_{i=1}^{n}\left(x_i,y_{i+1} - x_{i+1},y_i\right)\right\|.

## Examples Of The Area Of Polygons

### Example 1

**Find the area of a regular hexagon with side 6 cm and apothem 5.2 cm.**

Perimeter P=6×6=36 cm. A=\frac{1}{2}(36)(5.2). A=18×5.2=93.6 cm².

### Example 2

**Find the area of an irregular quadrilateral plot from its side lengths alone: sides 5, 6, 7, 8 m.**

Wrong path first: Average side method gives incorrect area; triangulate with a known diagonal or use coordinates.

### Example 3

**Find the area of a triangle with vertices (1, 2), (4, 6), (7, 1) using the shoelace formula.**

A=\frac{1}{2}|24−51|=\frac{1}{2}(27)=13.5 square units.

### Example 4

**Find the area of a regular pentagon with side 8 cm, using the side-only formula.**

A=\dfrac{5 \times 64}{4 \times 0.7265} \approx 110.1 cm².

### Example 5

**Find the area of the quadrilateral with vertices A(1,1), B(4,1), C(5,3), D(2,4) by the shoelace formula.**

A=\frac{1}{2}|35−19|=8 square units.

### Example 6

**Find the area of an irregular pentagon by triangulation.**

A=12+9+15=36 cm².

## Common Mistakes When Finding Polygon Area

### Mistake 1: Using the regular-polygon formula on an irregular shape

**Correct way:** Confirm the polygon is regular before using the apothem formula.

### Mistake 2: Forgetting to wrap the last vertex back to the first in the shoelace formula

**Correct way:** Always include the wrap-around term connecting the last vertex to the first.

### Mistake 3: Listing the vertices out of order

**Correct way:** List the vertices strictly in sequence around the boundary.

## Conclusion

- The **area of a polygon** is the space it encloses, depending on whether the polygon is regular or irregular.
- A **regular polygon** uses A=\frac{1}{2}Pa, or the side-only form A=\dfrac{n s^2}{4\tan(180°/n)}.
- An **irregular polygon** uses **triangulation** or the **shoelace formula** from ordered coordinates.
- Side lengths alone never determine an irregular polygon's area.