Irregular Polygon Area Calculator — Shoelace Formula
Welcome to CalcuGo's Irregular Polygon Area Calculator! Using the Shoelace Formula (Gauss's area formula), it instantly computes the area and perimeter of any simple polygon from its vertex coordinates. Enter up to 10 vertices in counterclockwise or clockwise order, choose your unit system (metric or US imperial), and get the result in seconds.
Polygon Refresher — Basic Terminology
A polygon is a closed plane figure bounded by finitely many straight-line segments. Triangles and rectangles are simple examples; ellipses and crescents are not polygons.
- Simple polygon — edges never cross each other.
- Self-intersecting polygon — some edges cross. To find the area of such a polygon, split it into two or more simple polygons first.
- Regular polygon — convex, equilateral, and equiangular (e.g., a square). Regular polygons have a dedicated formula; for everything else, the shoelace formula is the universal tool.
How to Find the Area of an Irregular Polygon — Shoelace Theorem
The shoelace formula (also called the surveyor's formula or Gauss's area formula) works for any simple polygon. You only need the Cartesian coordinates of its vertices.
Suppose a simple polygon has vertices listed in order (clockwise or counterclockwise):
(x₁, y₁), (x₂, y₂), …, (xₙ, yₙ)
Then:
Area = ½ × |x₁y₂ − y₁x₂ + x₂y₃ − y₂x₃ + … + xₙy₁ − yₙx₁|
The |…| denotes the absolute value, which ensures a positive area regardless of vertex order direction.
Perimeter = sum of Euclidean distances between consecutive vertices (and the last vertex back to the first).
How to Use This Irregular Polygon Area Calculator
- Select your unit system — Metric (m, cm, mm, km) or US Imperial (ft, in, yd, mi).
- Enter the X and Y coordinates of each vertex in the table. You can use up to 10 vertices. Leave unused rows blank.
- Make sure you are dealing with a simple polygon (no self-intersections). If your polygon crosses itself, divide it into simple parts and sum their areas.
- Enter vertices in counterclockwise or clockwise order — either works. Mixing the order will give a wrong result.
- Click Calculate. The calculator returns the area (in unit²) and the perimeter (in the chosen unit), plus a vertex table and the full shoelace expansion.
If your polygon has more than 10 vertices, divide it into two or more polygons, compute each area separately, and add the results.
Why Is It Called the "Shoelace Formula"?
The name comes from the visual pattern that appears when you multiply the coordinates. List the vertices in a column and repeat the first vertex at the bottom:
x₁ y₁ x₂ y₂ x₃ y₃ … xₙ yₙ x₁ y₁
Draw diagonal lines between each pair of adjacent rows — one set going down-right (multiplied with a + sign) and another going down-left (multiplied with a − sign). The crossing pattern of these lines resembles shoelace eyelets — hence the name.
For those familiar with linear algebra: the area equals ½ times the sum of the 2×2 determinants of consecutive coordinate pairs:
Area = ½ × ( det[x₁,y₁; x₂,y₂] + det[x₂,y₂; x₃,y₃] + … + det[xₙ,yₙ; x₁,y₁] )
Example — 5-Vertex Irregular Pentagon
Consider a polygon with vertices (listed counterclockwise):
| Vertex | X (m) | Y (m) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 6 | 0 |
| 3 | 8 | 4 |
| 4 | 3 | 7 |
| 5 | −1 | 5 |
Shoelace sum:
(0×0 − 0×6) + (6×4 − 0×8) + (8×7 − 4×3) + (3×5 − 7×(−1)) + ((−1)×0 − 5×0) = 0 + 24 + 44 + 22 + 0 = 90
Area = ½ × |90| = 45 m²
Perimeter = 6 + √20 + √34 + √20 + √26 ≈ 25.87 m
The default values pre-loaded in the form use exactly this polygon — click Calculate to see the result instantly.
Supported Unit Systems
| System | Units | Area Output |
|---|---|---|
| Metric | m, cm, mm, km | m², cm², mm², km² |
| US Imperial | ft, in, yd, mi | ft², in², yd², mi² |
FAQs
- Can I use negative coordinates?
- Yes. The shoelace formula works for any coordinate values, including negative ones. Only the vertex order matters.
- What if my polygon has self-intersections?
- The shoelace formula gives an incorrect result for self-intersecting polygons. Split the polygon into two or more simple (non-self-intersecting) polygons, calculate each area separately, and add them together.
- Does vertex order (clockwise vs. counterclockwise) matter?
- Either order is fine — the formula uses absolute value, so the result is always positive. However, mixing the order (e.g., jumping between non-adjacent vertices) will give a wrong answer.
- My polygon has more than 10 vertices. What do I do?
- Divide the polygon into two or more parts, each with ≤ 10 vertices (sharing an edge). Calculate the area of each part and sum the results.
- Is the shoelace formula exact?
- Yes — it produces an exact result for any simple polygon defined by rational coordinates, subject only to floating-point rounding in practice.
- What is the difference between area and perimeter?
- Area measures the surface enclosed by the polygon (in unit²). Perimeter is the total length of all sides (in unit).