Polygon and Land Area from Coordinates Calculator

How is area calculated from coordinates?

The area of a polygon with known corner coordinates comes from the Gauss area formula, better known as the shoelace formula because of the criss-cross pattern of the products. Take the corners in order, multiply each x by the next corner's y and each y by the next corner's x, add up the differences and halve the total:

A = |(x₁ × y₂ − x₂ × y₁) + (x₂ × y₃ − x₃ × y₂) + …| / 2

After the last corner you wrap around to the first. The formula is the same for any number of corners, concave shapes included.

Example

For the triangle with corners (0, 0), (4, 0) and (0, 3), the differences are 0 × 0 − 4 × 0 = 0, 4 × 3 − 0 × 0 = 12 and 0 × 0 − 0 × 3 = 0. The total is 12 and the area is 12 / 2 = 6 square units. Base times height over two gives the same: 4 × 3 / 2 = 6.

Land and plot areas

Enter plot corners in metres and the area comes out in square metres. The calculator also gives it in decares and hectares: 1 decare is 1000 m² and 1 hectare is 10 000 m². A plot with corners (0, 0), (60, 0), (75, 40), (35, 65) and (−10, 35) metres covers 3875 m², or 0.3875 hectares.

Map coordinates can run into the millions. The calculator shifts the working to the first corner, so precision is kept with large numbers.

Why does corner order matter?

Corners must be entered in order around the boundary. If the order is mixed up the sides cross and the formula returns a meaningless value. The calculator detects a self crossing polygon and warns you. Clockwise or counterclockwise makes no difference to the result, and the direction is reported separately.

Perimeter, sides and centroid

Each side length is the distance between two corners and the perimeter is their sum. The centroid is the point where the shape would balance if it were cut from a uniform sheet.

Pasting a list and CSV

Instead of typing coordinates one by one, copy two columns from a spreadsheet into the paste box or load a CSV file. Results can be downloaded as CSV.

Limits

Sides are taken as straight. Latitude and longitude in degrees cannot be used directly. The result is for information: the official area is the one in the land registry and cadastre records.