Area Calculator
The area of any polygon from its vertices — the shoelace sum printed term by term, the signed orientation named, and no shape table in sight.
Area Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Simple polygons only — self-intersecting boundaries are not detected
- Planar coordinates; the ring closes automatically
In short: A polygon’s vertices carry its whole area. Walk the boundary once and accumulate xᵢᵧ₊₁ − xᵢ₊₁ᵧ at each step: for the triangle (0, 0), (4, 0), (4, 3) the terms are 0, 12 and 0, the sum is 12, and half of it is 6 — the same 6 Heron gives on the triangle page for sides 3-4-5, computed with no side length in sight. The sign is information: a positive sum means the vertices ran counterclockwise; enter them clockwise and the raw sum comes out negative, which is the same area pointing the other way.
Formula
A = ½ · |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|
each edge contributes a signed rectangle strip — the shoelace laces cross, the areas cancel outside, and only the polygon survives.
Worked Example
- Close the ring. the last vertex laces back to the first automatically. If you repeated the first vertex to “close” the list, the page drops the duplicate and says so.
- Lace. at each vertex: xᵢ times the NEXT y, minus the next x times this y. Every term is printed.
- Halve the absolute sum. the sign survives to tell you the walking direction; the area is its absolute value.
(0, 0); (4, 0); (4, 3): terms 0·0 − 4·0 = 0, 4·3 − 4·0 = 12, 4·0 − 0·3 = 0. Sum 12, area 6. Reverse the order and the sum reads −12 — clockwise, same square feet.
Strengths & Limits Of This Model
Where this engine is strong
- Every lace term printed — the sum is auditable
- Winding direction named, not discarded
Where it stops
- No self-intersection detection (stated, not faked)
- No holes / multi-ring polygons
Practical Use Cases
Survey plots
a boundary walk from a plat or GPS pole list to area
Game maps and CAD
any simple polygon, convex or not
Homework
every cross-lace term printed, no black box
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Area Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why is it called the shoelace formula?
Write the coordinates in two columns and draw the cross multiplications: each x laces to the next y and each next x laces back to the current y. The crisscross pattern looks like lacing a shoe. The difference of the two laces, halved, is the area.
How can vertices alone determine area?
A simple polygon is exactly its boundary, and the boundary is exactly its vertex list in order. Between consecutive vertices the edge contributes a signed strip under the straight line; parts outside the polygon cancel against other edges’ negative strips. The shoelace sum is that cancellation done all at once.
What does the sign of the sum mean?
Direction. Positive means the vertices ran counterclockwise (the standard positive orientation); negative means clockwise. The area is the absolute value either way, but the sign is printed because graphics and survey code often need to KNOW the winding, not just the area.
Must the polygon be convex?
No — concave simple polygons work perfectly. The one honest restriction is SIMPLICITY: edges may not cross. The formula has no cheap way to detect a self-crossing, so the page states the assumption instead of pretending to check it.
Why is there no list of shape formulas here?
Because named shapes already own their questions: three sides live on the triangle page, one radius on the circle page, solids on the volume page. This page owns the one thing those cannot do — an arbitrary polygon whose only description IS its vertices.
What happens if I repeat the first vertex at the end?
The ring closes automatically, so the duplicate is dropped and the page says it dropped one. Keeping it would lace a zero-length edge and add a 0 term — harmless there, but the note keeps the working honest.
How many vertices can it take?
Up to 24 — far past any hand-drawn plot. The sum is linear in the vertex count, so even survey-grade boundary walks are instant.
Does it handle decimal or negative coordinates?
Freely. Signed coordinates are ordinary points — a polygon can live entirely in negative territory, and the lace algebra does not care. Only the count, the pairing and the simplicity of the boundary matter.