Using Determinants to find the Area of a Polygon

Using Determinants to Find the Area of a Polygon
The determinant method (commonly called the shoelace formula or Gauss’s area formula) provides an efficient way to calculate the area of any simple polygon when the coordinates of its vertices are known. This method works for both convex and concave polygons, as long as they don’t intersect themselves.

The following diagram shows how to find the area of a triangle using determinants. Scroll down the page for more examples and solutions.

 

Finding the area of a polygon using determinants
This video shows how to find the area of a polygon using determinants.

Determinants to Find the Area of a Polygon
This video shows how to use determinants to find the area enclosed by any polygon.

Common Mistakes to Avoid

1. Incorrect Vertex Order:
   Points must be ordered sequentially (clockwise or counter-clockwise).
   Plot the points first to verify order.
   
2. Missing the First Point at the End:
   The last term must loop back to (x₁,y₁)

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