Enter the coordinates of the vertices of a polygon in the form (x1,y1,x2,y2,…) into the calculator to determine the area of the polygon using the Shoelace Formula.

Shoelace Formula Calculator

Enter polygon vertex coordinates to compute area using the shoelace formula, or enter an area value to convert between units.

Area (selected unit):
Area (m²):
Area (km²):
Area (ft²):
Area (yd²):
Area (acres):
Perimeter (coordinate units):
Signed area (m²):
Orientation:
Centroid (x̄,ȳ):

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Shoelace Formula

The Shoelace Formula (also known as Gauss's area formula) is used to calculate the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. The formula is given by:

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

Variables:

  • A is the area of the polygon
  • (x_i, y_i) are the coordinates of the i-th vertex of the polygon
  • (x_(i+1), y_(i+1)) are the coordinates of the (i+1)-th vertex of the polygon

To calculate the area of a polygon, apply the Shoelace Formula by summing the products of the x-coordinates of each vertex by the y-coordinate of the next vertex, subtracting the sum of the products of the y-coordinates of each vertex by the x-coordinate of the next vertex, and then dividing the absolute value of the result by 2.

What is the Shoelace Formula?

The Shoelace Formula is a mathematical algorithm that provides a simple way to find the area of a polygon when the coordinates of its vertices are known. It is particularly useful because it does not require breaking the polygon into triangles and summing their areas; instead, it uses a direct calculation that involves the coordinates of the vertices in a specific order.

How to Calculate Area with the Shoelace Formula?

The following steps outline how to calculate the area of a polygon using the Shoelace Formula.

  1. List the coordinates of the vertices of the polygon in clockwise or counterclockwise order.
  2. Use the formula A = |(Σ(x_i * y_(i+1)) - Σ(y_i * x_(i+1)))| / 2 to calculate the area.
  3. Enter the coordinates into the calculator above to verify your result.

Example Problem :

Use the following coordinates as an example problem to test your knowledge.

Coordinates of the vertices of the polygon: (3,4), (5,11), (12,8), (9,5), (5,6)

Enter the coordinates in the form: 3,4,5,11,12,8,9,5,5,6