Shoelace Formula Calculator
Calculate polygon area from vertex coordinates with step-by-step calculations
Input Coordinates
Enter one point per line as x,y. Points should be in order (clockwise or counterclockwise).
Polygon Visualization
Result
Formula
A = ½|Σ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|
Also known as the surveyor's formula or Gauss's area formula
Understanding the Shoelace Formula
What is the Shoelace Formula?
The shoelace formula (also called the surveyor's formula or Gauss's area formula) calculates the area of any simple polygon given the coordinates of its vertices. It's called "shoelace" because the calculation pattern resembles the criss-cross pattern of shoelaces.
The Formula
For vertices (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ):
A = ½|x₁y₂ + x₂y₃ + ... + xₙy₁ - (x₂y₁ + x₃y₂ + ... + x₁yₙ)|
Or more compactly: A = ½|Σ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|
How to Use It
- List all vertices in order (clockwise or counterclockwise)
- For each pair of consecutive vertices, calculate xᵢ × yᵢ₊₁ and xᵢ₊₁ × yᵢ
- Sum all the first products (xᵢ × yᵢ₊₁)
- Sum all the second products (xᵢ₊₁ × yᵢ)
- Subtract the second sum from the first
- Take the absolute value and divide by 2
Why Does It Work?
The formula is derived from Green's theorem in vector calculus. Geometrically, it sums the signed areas of trapezoids formed between the polygon and the x-axis. The criss-cross pattern ensures that areas outside the polygon cancel out, leaving only the polygon's area.
Advantages
- Works for any simple polygon (convex or concave)
- Doesn't require the polygon to be regular
- Only needs vertex coordinates, no angles or side lengths
- Computationally efficient for computer algorithms
- Used extensively in surveying, GIS, and computer graphics
Frequently Asked Questions
Why is it called the shoelace formula?
The name comes from the visual pattern of the calculation. When you write out the coordinates and draw lines connecting the products (xᵢyᵢ₊₁ and xᵢ₊₁yᵢ), the pattern looks like shoelaces being laced through eyelets.
Does the order of vertices matter?
The vertices must be in order (either all clockwise or all counterclockwise around the polygon). The absolute value in the formula ensures the area is positive regardless of which direction you choose.
Can I use this for polygons with holes?
Yes, but you need to be careful. Calculate the outer polygon area, then calculate the hole area separately and subtract it. Make sure the outer polygon vertices go in one direction and the hole vertices go in the opposite direction.
What if I get a negative area?
The formula can give a negative result if vertices are in a certain order, but we take the absolute value, so the final area is always positive. The sign indicates whether you went clockwise or counterclockwise.
Does this work for self-intersecting polygons?
The shoelace formula technically works for self-intersecting polygons, but the result may not be what you expect. It calculates a "signed area" where some regions may cancel out. It's best used for simple (non-self-intersecting) polygons.
Who discovered this formula?
The formula is attributed to Carl Friedrich Gauss, though similar methods were known earlier. It's been used by surveyors for centuries to calculate land areas from field measurements.
Can I use this in programming?
Yes! The shoelace formula is perfect for programming because it's simple to implement with a loop. It's widely used in computational geometry, GIS software, CAD programs, and game engines.
What about 3D polygons?
The shoelace formula is for 2D polygons. For 3D polygons (planar polygons in 3D space), you need to project them onto a plane first or use the cross product method to calculate area.