How this calculator works
The shoelace sum adds signed triangle contributions around the boundary. Taking its absolute value gives the enclosed area; summing edge lengths gives perimeter. Concave outlines are allowed. Crossings, repeated vertices and overlapping edges are rejected because they do not describe one simple boundary. Holes and geographic latitude-longitude coordinates are outside this model. The tool returns numbers rather than a polygon drawing.
Formula
Area = (1/2)|sum(xi y(i+1) - x(i+1) yi)|, closing the last vertex back to the first. Perimeter = sum hypot(x(i+1)-xi, y(i+1)-yi).
Worked example
The ordered rectangle (0,0), (4,0), (4,3), (0,3) has area 12 and perimeter 14. Reversing the order changes its signed area from 12 to -12 without changing its size.
Advertisement
Frequently asked questions
Can the polygon be concave?
Yes, provided its boundary does not cross or touch itself.
Should I repeat the first point?
No. The closing edge is added automatically.
What units does area use?
If coordinates use metres, area is in square metres and perimeter in metres. Use the same unit on both axes.