Polygon Area from Coordinates (Shoelace Formula)
Enter vertex coordinates in order to compute a polygon area with the shoelace formula. Also returns vertex count, perimeter, and centroid, handles concave shapes, and draws a coordinate diagram.
Last updated:
Input
Enter the vertex coordinates one point per line, in order around the polygon. The area is computed with the shoelace formula, along with the perimeter and centroid.
Enter x and y separated by a comma on each line. At least 3 vertices are required.
Result
Area
16
Vertices
5
Perimeter
15.656854
Centroid
(2, 2.041667)
The area is computed with the shoelace formula (Gauss area formula). It assumes a simple polygon without self intersections and supports concave shapes.
How it works
- The area is computed with the shoelace formula (Gauss area formula). The signed area A sums x_i times y_next minus x_next times y_i over every edge, then divides by 2.
- The displayed area is the absolute value of this signed area. Listing vertices counter clockwise gives a positive sign and clockwise gives a negative sign, but the area value does not depend on direction.
- The perimeter sums the distance between consecutive vertices across all edges, including the closing edge from the last vertex back to the first, treating the polygon as closed.
- The centroid is found with an area weighted formula. For a degenerate shape whose area is near zero, the plain average of the vertex coordinates is used instead.
- A simple polygon without self intersections is assumed. Concave shapes are supported, but figures whose edges cross each other may not give a correct area.
- Each line holds x and y separated by a comma. Blank lines are ignored, and at least 3 vertices are required.
Reviews
Tell us what you think of this calculator.
Write a review
Found an incorrect result or a problem? Turn this on to report it.
Rating