What does this calculator measure?
The shoelace formula finds the signed area enclosed by an ordered simple polygon. This tool accepts convex or concave straight-edged boundaries and keeps coordinate arithmetic exact with scaled integers. The separate perimeter calculation uses Euclidean edge lengths.
For related tasks, use the existing Circle Calculator or Polygon Area & Perimeter Calculator.
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Optional geometry sets (paid link) can help draw a manual dimension sketch. No purchase is required.
How to enter values and calculate
Enter one x, y pair per line, such as 0, 0 then 4, 0 then 4, 3. Parentheses or a space separator are accepted. Follow the boundary in order. You may repeat the first point once at the end; closure is otherwise automatic. Select Calculate after completing all points.
The result and calculated drawing clear whenever an input changes. Reset returns the form to its initial state. Copy, Download and Print retain the result and calculation steps.
Optional graph paper notebooks (paid link) can help organize written working.
Formula and calculation method
Signed area = ½ Σ(xi yi+1 − xi+1 yi), with the final vertex paired with the first. Area is the absolute signed area. Perimeter is Σ hypot(xi+1 − xi, yi+1 − yi). Positive signed area indicates counterclockwise order; negative indicates clockwise order.
Steps are generated from the actual inputs. Unsupported input receives an error instead of a guessed result.
Worked example and interpretation
For (0,0), (4,0), (4,3), (0,3), the shoelace sum is 24, giving area 12 square units. Edge lengths 4,3,4,3 give perimeter 14 units. Reversing the order changes orientation but leaves area and perimeter unchanged. A concave boundary (0,0), (4,0), (4,4), (2,2), (0,4) has area 12 square units.
Use the same inputs on the calculator to reproduce these values and inspect the diagram.
Understand the result and diagram
P1, P2 and later labels correspond to the entered lines after removing an optional closing duplicate. The plot closes Pn to P1. It fits the actual boundary uniformly; it does not stretch x and y independently. Dense labels or extremely thin edges may overlap at screen resolution. Read the entered coordinates and edge steps for precise values.
Optional protractors (paid link) can help with angle conventions and geometry practice. For two Cartesian points, the existing Distance & Midpoint Calculator provides a separate workflow.
Accuracy, input limits and assumptions
Use 3–100 distinct vertices and ordinary decimal coordinates between −1 billion and +1 billion, with at most six decimal places. Exponent notation and fractions are not supported. Crossings, nonadjacent touches, duplicate vertices, overlapping edges and zero-area polygons are rejected using exact integer predicates. Holes, latitude/longitude, unordered-point sorting and curved boundaries are unsupported.
Optional geometry workbooks (paid link) can support additional practice. No measurement uncertainty, material allowance or construction design is inferred.
Polygon Area from Coordinates Calculator FAQs
Enter vertices consecutively around the boundary, clockwise or counterclockwise. The tool does not sort unordered points.
Yes, provided the boundary is simple and has no crossings, touching nonadjacent edges or overlapping edges.
Yes, for the entered ordinary decimal coordinates with at most six decimal places. Perimeter uses approximate square roots.
It is optional. One closing copy of the first vertex is removed before calculation; all other duplicate vertices are rejected.
No. Use Cartesian x, y coordinates in one shared linear unit. Geographic area, holes and curved edges are outside scope.
Yes. It connects the entered vertices in order, labels P1 through Pn and uses equal x/y scale. Very thin shapes may collapse to a line at screen resolution.
Sources and methodology
Independently implemented standard geometry formulas, reviewed October 29, 2026. Sketches explain dimension conventions. Sources describe the mathematics; the linked organizations do not endorse this calculator.