How to use the polygon area from coordinates calculator
- List the corner points of the shape, one per line as x, y, going around the edge in order. Either direction works.
- Add an optional unit label such as ft or m. It is only printed after the results.
- Read the area, the perimeter, the direction of the points and the centroid.
- If the page warns that edges cross, reorder the points so the outline does not cross itself.
Formula
signed area = (1/2) x sum of (x1 y2 - x2 y1) over every edge, including the closing edge
area = absolute value of the signed area
perimeter = sum of the edge lengths
centroid = sum of (x1 + x2)(x1 y2 - x2 y1) / (6 x signed area), and the same with y
This is the shoelace formula. Each edge adds the signed area of a thin triangle from the origin, and triangles outside the shape cancel, leaving the area of the shape.
A six sided lot
Take the corners (0, 0), (12, 0), (12, 5), (8, 9), (3, 9) and (0, 6) in feet. The shoelace terms x1 y2 - x2 y1 for the six edges are 0, 60, 68, 45, 18 and 0, which add to 191. Half of that is 95.5 square feet.
The edges are 12, 5, 5.657, 5, 4.243 and 6 feet, so the perimeter is 37.8995 feet. The centroid is at (5.8447, 4.0698). Splitting the shape into triangles from the first corner and adding their areas gives the same 95.5. A triangle with corners (0, 0), (4, 0) and (0, 3) has area 6, and going the other way around gives the same area with the direction shown as clockwise.
Entering the points
Walk around the shape and list each corner once. Do not repeat the first corner at the end, since the tool closes the shape for you. Use the same unit for x and y. The result is in squares of that unit.
For a land or building plan, use coordinates measured on a flat plane, such as survey coordinates in a projected system, and not latitude and longitude. A licensed surveyor should confirm any boundary. For simple shapes, the Area Calculator works from side lengths.
Edges that cross
If two edges cross, such as a bow tie shape, the shoelace formula returns a net area in which parts that wind in opposite directions cancel. The tool warns when this happens, because the number is not the area of any simple region. Reordering the points so the outline does not cross itself fixes it.
More shape tools
For other shape and measurement questions, try the Longest Length Around a Corner Calculator, Bike Spoke Length Calculator and IBU Calculator.
Assumptions and limits
- The points lie on a flat plane and are listed in order around the boundary.
- The shape is closed by joining the last point to the first.
- The centroid is shown only when the outline does not cross itself.
- Curved edges are treated as straight lines between the points.
Frequently asked questions
How do I find the area of an irregular polygon from coordinates?
Use the shoelace formula: add x1 times y2 minus x2 times y1 for every edge, including the closing one, and halve the absolute value. The tool shows each term.
Does the order of the points matter?
Yes. They must go around the boundary in order, clockwise or counterclockwise. Listing them out of order makes edges cross and gives a wrong area.
What is the centroid of a polygon?
It is the center of balance of the flat shape, the average position of all its area. For the lot above it sits at about (5.84, 4.07).
Can I use latitude and longitude?
Not directly. Degrees of longitude shrink away from the equator, so the formula would be wrong. Convert to flat coordinates in a map projection first.