Irregular Polygon Area Calculator

Calculate the area of any irregular polygon from vertex coordinates using the Shoelace formula.

Enter Vertex Coordinates

Enter the (x, y) coordinates of each vertex in order (clockwise or counterclockwise).

Result

Area
12
square units
Number of Vertices 4
Perimeter 14
Shoelace Sum 24

Step-by-Step Solution (Shoelace Formula)

A = |sum| / 2

Understanding the Shoelace Formula

The Shoelace formula (also known as Gauss's area formula or the surveyor's formula) is an elegant mathematical method for calculating the area of any simple polygon when you know the coordinates of its vertices. The name comes from the cross-multiplication pattern that resembles lacing a shoe.

Given n vertices (x1,y1), (x2,y2), ..., (xn,yn) listed in order, the area is:

Shoelace Formula

Sum the cross products of consecutive vertex coordinates and take half the absolute value.

A = |Σ(x_i * y_{i+1} - x_{i+1} * y_i)| / 2

How It Works

Multiply each x by the next y, subtract each next x times current y, sum all, take half absolute value.

Cross-multiply in a "shoelace" pattern

Vertex Order

Vertices must be in order (CW or CCW). The formula handles both directions via the absolute value.

Works for CW and CCW ordering

How to Use This Calculator

  1. Enter the (x, y) coordinates of each vertex in sequential order (clockwise or counterclockwise).
  2. Use "Add Vertex" or "Remove Vertex" to adjust the number of vertices.
  3. Click "Calculate Area" to see the result and step-by-step Shoelace calculation.
  4. The calculator also computes the perimeter of the polygon.

Requirements for Accurate Results

  • The polygon must be simple (non-self-intersecting).
  • Vertices must be listed in sequential order around the polygon.
  • At least 3 vertices are required to form a polygon.
  • The formula automatically closes the polygon by connecting the last vertex to the first.

Applications

  • Land surveying: Calculating plot areas from GPS coordinates.
  • Computer graphics: Determining polygon areas for rendering.
  • Geographic Information Systems (GIS): Computing areas of map regions.
  • Architecture: Calculating floor plan areas of irregular rooms.
  • Engineering: Cross-sectional area calculations for structural analysis.

Mathematical Background

The Shoelace formula is derived from the concept of signed areas. Each pair of consecutive vertices contributes a signed area based on the trapezoid formed between the edge and the x-axis. The sum of these signed areas gives the total signed area, and taking the absolute value gives the geometric area regardless of vertex ordering direction.

The formula is closely related to Green's theorem from vector calculus, which converts a line integral around a closed curve into a double integral over the enclosed region. The Shoelace formula is essentially a discrete version of this theorem applied to polygons.