KEVOS® MAKEOS · Geometry
Polygon Area and Centroid from Coordinates
What is the area, perimeter and centroid of this polygon from its corner coordinates?
Area (shoelace formula), perimeter and centroid of any simple polygon with up to eight corners, including irregular quadrilaterals (trapeziums) and plate outlines.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Number of corners | corners | How many of the listed corners are used (the first n), in order around the outline. |
| Corner 1 | ||
| X | mm | X coordinate of corner 1. |
| Y | mm | Y coordinate of corner 1. |
| Corner 2 | ||
| X | mm | X coordinate of corner 2. |
| Y | mm | Y coordinate of corner 2. |
| Corner 3 | ||
| X | mm | X coordinate of corner 3. |
| Y | mm | Y coordinate of corner 3. |
| Corner 4 | ||
| X | mm | X coordinate of corner 4. |
| Y | mm | Y coordinate of corner 4. |
| Corner 5 | ||
| X | mm | X coordinate of corner 5. |
| Y | mm | Y coordinate of corner 5. |
| Corner 6 | ||
| X | mm | X coordinate of corner 6. |
| Y | mm | Y coordinate of corner 6. |
| Corner 7 | ||
| X | mm | X coordinate of corner 7. |
| Y | mm | Y coordinate of corner 7. |
| Corner 8 | ||
| X | mm | X coordinate of corner 8. |
| Y | mm | Y coordinate of corner 8. |
Method and formulas
- Area = ½ |Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)| (shoelace formula).
- Centroid x̄ = Σ (xᵢ + xᵢ₊₁)(xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ) ÷ 6A, and similarly for ȳ (signed A).
- Perimeter = sum of side lengths.
- Example: the default quadrilateral has an area of 6,350 mm² with its centroid at (60.919, 28.976).
Limitations
- Simple polygons only: sides must not cross each other.
- Straight sides; arcs need their own segment areas added or subtracted.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the area and centroid of a plate or gusset
- Calculate the area of an irregular quadrilateral
- Check CAD areas
Who it is for
- Design engineers
- Toolmakers
- Estimators
What you get
- Area
- Perimeter
- Centroid
- Direction of the corner order
Interpreting the result
- List corners consecutively around the outline, either direction.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 40–48, 66–76. Topic reference; no source tables or recommended settings are reproduced.
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