Irregular Polygon Area Calculator
Calculate area and perimeter of irregular polygons using coordinates and the Shoelace formula
Polygon Coordinates
Vertices (4)
Enter vertices in clockwise or counter-clockwise order. Maximum 30 vertices.
Calculation Results
Centroid
x: 2.0000, y: 1.5000
Polygon Type
✅ Simple (non-self-intersecting)
Formula used: Shoelace formula: Area = ½|∑(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|
Vertices count: 4 points
Shoelace Formula Steps
1. List vertices in order: (0, 0), (4, 0), (4, 3), (0, 3)
2. Apply formula: Area = ½|∑(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|
3. Calculate cross products and sum them
4. Take absolute value and divide by 2
5. Result: 12.0000 square units
Example: Pacman Shape
Vertices
(0, -2), (6, -2), (9, -0.5), (6, 2), (9, 4.5), (4, 7), (-1, 6), (-3, 3)
Calculation
Cross products: 0×(-2) - (-2)×6 = 12
6×(-0.5) - (-2)×9 = 15
9×2 - (-0.5)×6 = 21
... (continue for all vertices)
Sum = 154, Area = 154/2 = 77 square units
Shoelace Formula
Where:
• (xᵢ, yᵢ) are vertex coordinates
• Sum over all vertices
• Take absolute value
• Divide by 2
Polygon Properties
Simple: No self-intersecting edges
Vertices must be in order (clockwise or counter-clockwise)
Minimum 3 vertices required
Maximum 30 vertices supported
Quick Tips
Enter vertices in clockwise or counter-clockwise order
Works with any simple polygon shape
Automatically detects self-intersections
Calculates both area and perimeter
Understanding the Shoelace Formula
What is the Shoelace Formula?
The Shoelace formula (also known as Gauss's area formula) is a mathematical method for calculating the area of a simple polygon when you know the coordinates of its vertices. It's called "shoelace" because the calculation pattern resembles the crisscross pattern of shoelaces.
Why Use This Method?
- •Works for any simple polygon shape
- •Only requires vertex coordinates
- •Fast and efficient calculation
- •Handles both convex and concave polygons
Step-by-Step Process
Step 1: List Vertices
List all vertices in order (clockwise or counter-clockwise)
Step 2: Apply Formula
For each vertex pair, calculate xᵢyᵢ₊₁ - xᵢ₊₁yᵢ
Step 3: Sum and Finalize
Sum all products, take absolute value, divide by 2
Important: Vertices must be listed in order around the polygon. Random order will give incorrect results!