Learn: Polygon area — shoelace
A boundary described by ordered survey coordinates encloses an area. The shoelace method adds signed cross-products of neighboring vertices, closes the final edge, and takes half the absolute total.
What the formula is saying
List each point as x,y in boundary order. For every edge from point i to the next point, calculate xᵢyᵢ₊₁ − xᵢ₊₁yᵢ. Include the last-to-first edge. Add the contributions before taking the absolute value and dividing by 2; individual negative terms must not be made positive.
Read the symbols in plain language
- coordinates
- Coordinates x,y — one point per line
Ordered planar boundary vertices. Enter x,y in metres, one pair per line; close the last edge to the first, without crossing edges or holes.
coordinates in mUse m coordinates as the base unit shown here. Use projected planar x and y coordinates in m, one point per line with a comma separating coordinates. The area is m². Latitude and longitude in degrees are not valid metre coordinates for this formula.
- A
- Result to find
Polygon area — shoelace. Only the total is made positive; negative edge contributions are required for correct concave-polygon area.
m²
Sort out the units first
Use projected planar x and y coordinates in m, one point per line with a comma separating coordinates. The area is m². Latitude and longitude in degrees are not valid metre coordinates for this formula.
Assumptions before calculating
Assume a simple non-self-intersecting polygon with at least three distinct vertices listed clockwise or counterclockwise. Concave polygons are allowed. The calculator accepts up to 200 vertices and automatically closes the boundary.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
A rectangular parcel is given by four ordered corners. Calculate its horizontal area and check against length times width.
- coordinates · Coordinates x,y — one point per line
0,0 10,0 10,5 0,5
coordinates in m
Follow every boundary edge
Subtract the first point from every vertex before multiplying. This translation preserves area and reduces cancellation with large map coordinates. Include the closing edge; take the absolute value only after summing.
| Edge | (xᵢ, yᵢ) | (xᵢ₊₁, yᵢ₊₁) | xᵢyᵢ₊₁ − xᵢ₊₁yᵢ (m²) |
|---|---|---|---|
| 1 → 2 | (0, 0) | (10, 0) | 0 |
| 2 → 3 | (10, 0) | (10, 5) | 50 |
| 3 → 4 | (10, 5) | (0, 5) | 50 |
| 4 → 1 | (0, 5) | (0, 0) | 0 |
Add the signed contribution of every boundary edge
The coordinate table shows each cross-product, including the final edge back to the first point.
(100) = 100 m²Take half the absolute signed sum
Only the total is made positive; negative edge contributions are required for correct concave-polygon area.
abs((100)) ÷ 2 = 50 m²
Does this worked answer make sense?
Reversing the entire boundary order changes the signed sum but not the final area. Translating all points by the same offset also leaves area unchanged; the implementation shifts the origin internally to reduce cancellation.
A second worked example — different values
An L-shaped parcel is represented by six ordered corners. Calculate its area without filling in the missing rectangular corner.
- coordinates · Coordinates x,y — one point per line
0,0 6,0 6,2 2,2 2,5 0,5
coordinates in m
Follow every boundary edge
Subtract the first point from every vertex before multiplying. This translation preserves area and reduces cancellation with large map coordinates. Include the closing edge; take the absolute value only after summing.
| Edge | (xᵢ, yᵢ) | (xᵢ₊₁, yᵢ₊₁) | xᵢyᵢ₊₁ − xᵢ₊₁yᵢ (m²) |
|---|---|---|---|
| 1 → 2 | (0, 0) | (6, 0) | 0 |
| 2 → 3 | (6, 0) | (6, 2) | 12 |
| 3 → 4 | (6, 2) | (2, 2) | 8 |
| 4 → 5 | (2, 2) | (2, 5) | 6 |
| 5 → 6 | (2, 5) | (0, 5) | 10 |
| 6 → 1 | (0, 5) | (0, 0) | 0 |
Add the signed contribution of every boundary edge
The coordinate table shows each cross-product, including the final edge back to the first point.
(36) = 36 m²Take half the absolute signed sum
Only the total is made positive; negative edge contributions are required for correct concave-polygon area.
abs((36)) ÷ 2 = 18 m²
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
A triangular parcel uses a different set of three corners. Find its plan area and compare your result with half base times perpendicular height.
- coordinates · Coordinates x,y — one point per line
0,0 8,0 3,4
coordinates in m
Find: Learn: Polygon area — shoelace
A hint, not the answer
List each point as x,y in boundary order. For every edge from point i to the next point, calculate xᵢyᵢ₊₁ − xᵢ₊₁yᵢ. Include the last-to-first edge. Add the contributions before taking the absolute value and dividing by 2; individual negative terms must not be made positive.
Use projected planar x and y coordinates in m, one point per line with a comma separating coordinates. The area is m². Latitude and longitude in degrees are not valid metre coordinates for this formula.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Follow every boundary edge
Subtract the first point from every vertex before multiplying. This translation preserves area and reduces cancellation with large map coordinates. Include the closing edge; take the absolute value only after summing.
| Edge | (xᵢ, yᵢ) | (xᵢ₊₁, yᵢ₊₁) | xᵢyᵢ₊₁ − xᵢ₊₁yᵢ (m²) |
|---|---|---|---|
| 1 → 2 | (0, 0) | (8, 0) | 0 |
| 2 → 3 | (8, 0) | (3, 4) | 32 |
| 3 → 1 | (3, 4) | (0, 0) | 0 |
Add the signed contribution of every boundary edge
The coordinate table shows each cross-product, including the final edge back to the first point.
(32) = 32 m²Take half the absolute signed sum
Only the total is made positive; negative edge contributions are required for correct concave-polygon area.
abs((32)) ÷ 2 = 16 m²
Avoid the common trap
Do not sort points merely by their x coordinate, omit the closing edge, or take the absolute value of every edge contribution. Use a comma as coordinate separator, not as a thousands separator or decimal mark.
When this method applies — and when it does not
Holes and multiple separate parcels require separate signed-area treatment and are not encoded by a single boundary here. The result is a horizontal plan area, not sloping surface area or a cadastral approval. Degenerate, repeated and crossing boundaries are rejected.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Polygon area — shoelace. Survey coordinate and traverse computations: specify the azimuth origin, rotation direction and planar coordinate units before calculating components or area.
- U.S. Army Corps of Engineers — Control and Topographic Surveying, EM 1110-1-1005 (2007)
- MIT OpenCourseWare — Multivariable Calculus: Green’s theorem and area (18.02SC, 2010)
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.