Polygon Area from Coordinates
Calculate area, perimeter and area centroid from 3–50 ordered vertices of a simple polygon. Inspect an equal-scale outline and a table of edges.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Join vertices in the entered boundary order and close the last edge to the first point. The shoelace sum gives signed area; its absolute half is the area. Sum Euclidean edge lengths for perimeter and use signed area moments for centroid.
A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|; P = Σ√[(Δx)²+(Δy)²]
Worked example
Vertices (0,0), (4,0), (4,3), (0,3) in meters give area 12 m², perimeter 14 m and centroid (2,1.5) m. Reversing the boundary order preserves these quantities.
Assumptions and input details
Use planar Cartesian coordinates in a shared unit, with one x,y pair per line or semicolon. Do not enter latitude/longitude. Convex and concave simple polygons are supported; holes, self-intersections, nonadjacent touching edges and overlapping edges are rejected. A repeated first vertex at the end is removed. Coordinates are limited to ±1e9 in the selected input unit. Computation translates and normalizes coordinates; geometry at or below a relative tolerance of 1e−12 is rejected as unresolved. The centroid assumes uniform area density and can lie outside a concave polygon.
Unit definitions
Unit selection is independent of language. Each selector contains only units for its physical quantity. Calculations use defined base units internally.
Choose from 23 length definitions, including explicitly labelled regional measures. Area conversions square the length factor; geometric volumes cube it.
Regional measures are not interchangeable merely because their names look similar. This is a documented selection, not a claim to include every historical unit.
NIST measurement tables · Japanese traditional units · Mainland China and Hong Kong unit distinctions
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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