Find the area of a polygon from its vertex coordinates with the shoelace formula, with each cross-multiplication step shown, plus perimeter and centroid.

Vertices in order around the shape, one x,y pair per line, or pasted as (x1,y1), (x2,y2), โ€ฆ. The last point joins back to the first.

A unit labels all coordinates; changing it reinterprets the numbers, rather than rescaling the point list.

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Shoelace Formula

The Shoelace Formula (also known as Gauss’s area formula) calculates the area of a simple planar polygon from its ordered Cartesian vertices. Follow the boundary without crossings, nonadjacent touches or overlaps; this calculator does not model holes, multiple contours or geographic latitude/longitude. The formula is given by:

A = |(ฮฃ(xi ร— y(i+1)) – ฮฃ(yi ร— x(i+1)))| / 2

Variables:

  • A is the area of the polygon
  • (xi, yi) are the coordinates of the i-th vertex of the polygon
  • (x(i+1), y(i+1)) are the coordinates of the next vertex; after the final vertex, use the first vertex to include the closing edge.

Sum xi ร— y(i+1) and subtract the sum of yi ร— x(i+1), including the final-to-first edge. Half this signed difference is positive for counterclockwise order and negative for clockwise order when x points right and y points up; screen y-down axes reverse that visual convention. Take the absolute value for area. The uniform filled-area centroid uses cross-weighted vertex sums divided by 6 times the signed area, not the average of the vertices; see the primary area and centroid formulas. Perimeter includes the closing edge. Area units use the square of the length conversion factor, while centroid and perimeter retain the coordinate length unit.

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 3โ€“200 distinct vertices in their boundary order. Enter one complete x,y or x y pair per line, with optional parentheses; do not use a flat comma list. One final repeat of the first point is optional and is normalized by the calculator. Forward collinear boundary vertices are allowed; backtracking and zero area reject.
  2. Use the formula A = |(ฮฃ(xi * y(i+1)) – ฮฃ(yi * 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 one pair per line:

3,4
5,11
12,8
9,5
5,6

The forward sum is 207 and the backward sum is 267, so the signed area is (207 โˆ’ 267)/2 = โˆ’30 and the area is 30 square units. The order is clockwise under Cartesian y-up axes. The filled-area centroid is (43/6, 137/18) โ‰ˆ (7.166667, 7.611111), and the perimeter is approximately 26.090056 units. If these coordinates are metres, the area is 30 mยฒ = 300,000 cmยฒ. These are planar point calculations, not estimates of measurement uncertainty.