FOOTPRINT GUIDE

Order the corners.
Close the boundary.

Plotarea measures one simple flat polygon from local x,y coordinates. “Simple” means its boundary does not cross or touch itself: consecutive edges meet only at their shared vertex. An inward corner is allowed.

Start with a local coordinate system

Choose one origin and perpendicular x and y directions, then express every corner in the same length unit. A corner to the left or below the origin may use a negative coordinate. The origin may be inside or outside the shape; moving the whole coordinate system does not change the footprint’s area or perimeter.

Enter one x,y pair per line, separated by a comma. Start at any corner and follow the actual boundary all the way around, clockwise or counterclockwise. Do not jump across the shape to list a more convenient point. The last vertex connects to the first automatically. If you prefer, repeat the first point once at the very end; that closing repeat is not counted as an extra vertex.

These are local planar coordinates. Latitude and longitude are angular geographic coordinates and are outside this tool. It does not project a map, interpret a cadastral plan, account for Earth curvature or measure a sloping surface. Use it for the mathematical outline represented by the entries, with measurement quality supplied by your own process.

Area from ordered cross-products

For adjacent corners (xᵢ,yᵢ) and (xᵢ₊₁,yᵢ₊₁), form xᵢyᵢ₊₁ − xᵢ₊₁yᵢ. Include the closing pair from the last corner back to the first. Add the terms, take the absolute value and divide by two:

This is the shoelace formula for the validated simple polygon. The sign before taking the absolute value records entered orientation. Reversing the order reverses that sign while keeping the unsigned area. A counterclockwise order has a positive signed sum in the usual x-right, y-up coordinate convention.

The formula can give a cancelling signed quantity for a crossing loop. That quantity is not the ordinary footprint area intended here. Plotarea validates the boundary before showing a usable area result, rather than interpreting a self-crossing drawing by a fill rule.

Worked rectangle: area 12, perimeter 14

Enter (0,0), (4,0), (4,3), (0,3) in metres. The four cross-product terms are 0, 12, 12 and 0. Their sum is 24, so the enclosed area is 12 m².

The closed edge lengths are 4, 3, 4 and 3 m. Adding all four gives a perimeter of 14 m. Omitting the closing edge would give an incomplete boundary length. Reversing the list or choosing a different first corner leaves both size results unchanged.

Worked concave L: area 6, perimeter 14

The default outline is (0,0), (4,0), (4,1), (1,1), (1,3), (0,3). It can be independently decomposed into a 4-by-1 m horizontal rectangle and a 1-by-2 m vertical extension: 4 + 2 = 6 m². Their interiors do not overlap.

The boundary lengths are 4, 1, 3, 2, 1 and 3 m, for a total of 14 m. A 4-by-3 m bounding rectangle has area 12 m², but its missing 3-by-2 m corner is not inside this outline. A concave inward turn is valid; the boundary still has one unambiguous interior.

Worked triangle: the diagonal is part of the perimeter

The vertices (0,0), (3,0), (0,4) enclose a right triangle of area 3 × 4 ÷ 2 = 6 square units. The diagonal from (3,0) to (0,4) is 5 units, so the boundary totals 3 + 5 + 4 = 12 units.

General diagonals involve square roots. Plotarea displays edge lengths and perimeter as numerical approximations, using up to 12 significant digits. It subtracts exact canonical coordinates before converting differences for those approximations, which avoids losing a small edge merely because the chosen origin uses large coordinates.

Which boundaries are rejected?

A bow tie through (0,0), (4,4), (0,4), (4,0) crosses itself. A notch corner placed exactly on a nonadjacent edge also touches the boundary and is rejected. Collinear overlapping edges, a repeated interior corner and an adjacent edge that reverses back along its predecessor are rejected too.

A forward intermediate point on a straight edge is allowed. For example, adding (2,0) between (0,0) and (4,0) preserves the rectangle’s area and perimeter. Turning back from (4,0) to (2,0) along an edge just travelled is different: it overlaps the boundary and cannot be used as a simple outline.

The checks use exact orientation signs and exact segment bounds for the entered decimal coordinates. There is no proximity tolerance that turns a genuine crossing into an acceptable one. A corner just above an edge is distinct from one exactly on it, even when their diagrams look similar. Software validation cannot establish the physical accuracy or uncertainty of the measurements.

Metres, international feet and square units

The international foot is exactly 0.3048 m. A square foot is therefore exactly 0.09290304 m². A 4-by-3 ft rectangle encloses 12 ft² = 1.11483648 m², and its 14 ft perimeter is 4.2672 m.

Changing the unit converts the displayed list and retains a canonical snapshot of the physical coordinates. It does not reinterpret the same numbers in a different unit. Converted coordinates round to at most 12 decimal places; a notice identifies that situation. Repeated unit changes keep the unedited physical shape and exact area intact.

Editing the list deliberately replaces every canonical coordinate with the displayed entries. This matters for a very small detail: a converted decimal view may round two nearly touching positions to the same visible number, while the unedited canonical geometry remains distinct. If you edit that list, the rounded visible geometry becomes the new input and may correctly fail the boundary check. The original text report retains exact fractional coordinates so the distinction is available.

What is exact, and what is rounded?

Supported input decimals are stored as exact fixed-scale integers in metres. Cross-products, area and boundary-intersection decisions use integer arithmetic. The exact enclosed area is shown as an integer or reduced fraction; a repeating square-foot result is not replaced by a rounded intermediate value.

The main area and coordinate views round to at most 12 decimal places. If a positive area is too small for a nonzero decimal at that precision, the display says it is below 0.000000000001 in the selected square unit and retains its exact fraction separately. This is not a zero-area result. Numerical perimeter and diagram pixels remain approximate views, and none of these arithmetic digits establishes real measurement accuracy.

Input limits and local reports

Use 3–100 distinct vertices, plus at most one repeated start point at the end. Input text is limited to 20,000 characters and each coordinate token to 32 characters, with up to 12 decimal places. Signed plain decimals such as −0.5, +1.25 and .5 are supported. Exponents, headings, extra columns and interior blank rows are not accepted. Each coordinate magnitude must be within 1,000,000,000 metres equivalent. These are software bounds.

Invalid entries hide the previous results, diagram and edge table and disable exports. A malformed coordinate list blocks a unit conversion until corrected. A syntactically valid but crossing outline can be converted between units, but it remains invalid. Reset restores the original six-vertex L-shape after the browser’s native reset completes.

Copy and Text include exact area, exact fractional coordinates in both the selected unit and metres, and approximate edge/perimeter values. CSV separates vertex, edge and summary records and labels precision. Import fraction fields as text if your spreadsheet would otherwise interpret them as dates. Reports are generated locally; they do not save a cloud project.

Primary mathematical context

MIT OpenCourseWare’s shoelace lecture (PDF) develops coordinate area and discusses why intersecting loops need a different interpretation. CGAL’s official polygon manual defines simple boundaries. Plotarea uses its own bounded implementation and does not load the CGAL library. NIST’s unit reference records the international-foot definition used here.

One flat outline is the full scope: no holes, curves, islands, automatic point sorting, slope correction, material-purchase calculation, legal-boundary or structural-design assessment.

Return to the footprint calculator →