BIZOMAT 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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Number of corners
- What it is
- How many corners of the outline the tool uses. It takes the first n corners in the list, in order, and closes the outline from the last back to the first.
- Units and format
- Enter a number in corners: corners. The form accepts 3 corners to 8 corners. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 corners as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Corners after the n-th are ignored, so leftover values there do not matter; list the corners in order around the outline.
Corner 1
What this means: X
- What it is
- Horizontal position of the first corner of the outline, measured from any origin you choose on the drawing or part.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use one origin and one axis direction for every corner; mixing datums gives a wrong area.
What this means: Y
- What it is
- Vertical position of the first corner of the outline, measured from the same origin as the X values.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Keep the Y axis pointing the same way for every corner.
Corner 2
What this means: X
- What it is
- Horizontal position of the second corner, the next one along the outline after corner 1, from the same origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 120 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Go round the outline in one direction; jumping across it makes the sides cross and the area wrong.
What this means: Y
- What it is
- Vertical position of the second corner, the next one along the outline after corner 1, from the same origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Read both coordinates of a corner from the same point on the drawing.
Corner 3
What this means: X
- What it is
- Horizontal position of the third corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A polygon needs at least three corners, so corner 3 is always used.
What this means: Y
- What it is
- Vertical position of the third corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 70 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A polygon needs at least three corners, so corner 3 is always used.
Corner 4
What this means: X
- What it is
- Horizontal position of the fourth corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 4 or more.
What this means: Y
- What it is
- Vertical position of the fourth corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 4 or more.
Corner 5
What this means: X
- What it is
- Horizontal position of the fifth corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 5 or more; otherwise ignored.
What this means: Y
- What it is
- Vertical position of the fifth corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 5 or more; otherwise ignored.
Corner 6
What this means: X
- What it is
- Horizontal position of the sixth corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 6 or more; otherwise ignored.
What this means: Y
- What it is
- Vertical position of the sixth corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 6 or more; otherwise ignored.
Corner 7
What this means: X
- What it is
- Horizontal position of the seventh corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 7 or 8; otherwise ignored.
What this means: Y
- What it is
- Vertical position of the seventh corner along the outline, from the same origin as the other corners.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 7 or 8; otherwise ignored.
Corner 8
What this means: X
- What it is
- Horizontal position of the eighth corner, the last one before the outline closes back to corner 1.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 8; otherwise ignored.
What this means: Y
- What it is
- Vertical position of the eighth corner, the last one before the outline closes back to corner 1.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only when the number of corners is 8; otherwise ignored.
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.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Areas, centroids and mass from a 2D profile
In Drafting Geometry by the Numbers
What it covers: Calculate a profile's area by the shoelace formula and by composite shapes, find its centroid, turn it into mass, and check the CAD's region properties before anyone relies on them.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the Polygon Area and Centroid from Coordinates calculator