Area Of Polygons - Definition, Formula, and Examples
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.