Area Calculator

Purpose and model boundary

This calculator measures a flat area that you draw: a floor, a room, a deck or a lot. You draw its outline on a grid and mark up to five areas inside it that do not count, such as columns, kitchen islands, cabinets, a planter or a building footprint. The spreadsheet returns the gross area inside the outline, the area of each exclusion, and the net area that remains. It measures plan area only. It does not estimate materials, waste, cost, wall or ceiling area, slopes, or volume.

Inputs and units

The drawing is the input. Its outline is sent as an ordered list of up to 100 corner points, and each exclusion as an ordered list of up to 20 points, one row per point with an x and a y coordinate. The outline closes itself from its last point back to its first, so the first point is never repeated.

The US/Metric switch on the drawing chooses the unit the coordinates are written in: feet for US and metres for Metric. The same switch sets the spreadsheet's US/Metric input, so the spreadsheet is told which unit it received. No formula reads that input; it names the unit of the results, which is square feet for US and square metres for Metric. Coordinates are written to 15 significant digits, the precision of a spreadsheet cell, so 85 inches is sent as 7.08333333333333 ft.

The drawing tool keeps its points in inches or centimetres, and switching between US and Metric converts them at exactly 2.54 cm per inch. The grid and the display unit change only how the drawing is shown and where a point snaps. They never change the coordinates that are sent, so changing them does not change a result.

Governing relationships

For a polygon with points (x₁, y₁) to (xₙ, yₙ) in drawing order, the area is the shoelace formula, also called the surveyor's area formula:

Area = ½ × |Σ (xᵢ₋₁ × yᵢ − xᵢ × yᵢ₋₁)| for i = 1 to n, where point 0 is point n

The term for the first point pairs it with the last point, which is what closes the outline. The formula is exact for any simple polygon, including angled walls; nothing is approximated.

Gross area = the area of the outline

Exclusion area k = the area of exclusion k, or 0 when it has fewer than three points

Net area = Gross area − (Exclusion area 1 + … + Exclusion area 5)

Calculation sequence

1. The spreadsheet counts the points in the outline block.
2. For each point it takes the previous point, which for the first point is the last one, and computes the cross product x_prev × y − x × y_prev. Rows after the last point contribute nothing.
3. Gross area is half the absolute value of the sum of those cross products, or 0 when the outline has fewer than three points.
4. Each of the five exclusion blocks is computed the same way to give its own area.
5. The five exclusion areas are added together.
6. Net area is the gross area less that total.

Outputs and interpretation

Net area is the headline result. Gross area is the whole area inside the outline, and Exclusion 1 area to Exclusion 5 area are the areas taken away, one per slot, 0 for a slot you did not use. All of them are shown to two decimals, in square feet for US and in square metres for Metric.

The drawing shows the dimensions of what you draw: the length of each wall of the outline and of each exclusion wherever its label fits without covering another label or a point, angled walls included, the clear distance from an exclusion to the walls around it while that exclusion is selected or being moved, and the length of a selected wall. These measure your drawing; they are not results, and they are how you draw an area to its measured size. The drawing never shows an area or a perimeter of its own. Its information panel repeats the spreadsheet's gross, net and exclusion areas only in the unit the spreadsheet returned them in, while the drawing's units are feet for US or metres for Metric; in inches, yards or centimetres it shows them as not available. The panel rounds a square-foot figure that is not whole to one decimal; the results below the drawing give two.

A result belongs to the drawing it was calculated for. When you change the drawing after pressing Calculate, the result is kept but marked stale until you press Calculate again; undoing back to the calculated drawing makes it current again. Switching between US and Metric changes the rows that are sent, so it marks the result stale too, and the next Calculate answers in the other unit.

Validation and status logic

The drawing tool checks every edit before it is kept. An edit that would break one of these rules is refused: the drawing goes back to how it was and the tool says why, in its own words.

• The outline and each exclusion have at least three points.
• No two consecutive points of a shape coincide.
• No wall crosses another wall of the same shape, so no shape turns inside out.
• Every exclusion lies inside the outline. Touching a wall is allowed.
• No two exclusions overlap. Touching is allowed, so no area is subtracted twice.
• There are at most five exclusions.

An imported drawing is checked the same way and refused as a whole if any rule fails. The drawing tool always writes a shape's points into consecutive rows, with both coordinates of every point; the spreadsheet refuses a request that leaves a blank row between them, or gives a point of the outline or of an exclusion only one coordinate.

The status line is the spreadsheet's own verdict on the rows it received, the first of these that applies:

• NOT VALID: choose US or Metric
• NOT VALID: draw at least three outline points
• NOT VALID: every outline point needs both coordinates
• NOT VALID: every point of exclusion 1 needs both coordinates
• NOT VALID: every point of exclusion 2 needs both coordinates
• NOT VALID: every point of exclusion 3 needs both coordinates
• NOT VALID: every point of exclusion 4 needs both coordinates
• NOT VALID: every point of exclusion 5 needs both coordinates
• NOT VALID: each exclusion needs zero or at least three complete points
• NOT VALID: outline and exclusion points must fill consecutive rows, with no blank row between them
• OK

A NOT VALID: verdict is shown in place of the figures, and the drawing's information panel then shows them as not available.

Assumptions and limitations

• The area is a plan area on a flat surface. A sloped roof, a ramp or uneven ground is measured as its footprint.
• Walls are straight lines between points. A curved wall has to be drawn as short straight segments, and segments whose ends lie on a curve cut inside it: a circle drawn as 32 such segments reports 0.64% less than the true circle.
• The spreadsheet trusts the drawing tool to keep exclusions inside the outline and apart from each other. It does not check either rule itself, so this calculator is offered on its web page only, where the drawing tool enforces them.
• There is no saved drawing between visits. Export saves the drawing as a file, and Import loads it back; the file carries a checksum of every coordinate, so a changed file is refused.
• Drawing needs a mouse, a pen or a touch screen. The drawing area cannot be operated with a keyboard or a screen reader in this version; a drawing prepared elsewhere can be loaded with Import.

Restrictions and non-computing states

This calculator accepts an outline of three or more points and up to five exclusions, each of three or more points, within the capacity of the rows it sends. The drawing tool refuses any edit that would leave a shape outside those limits, so every drawing on the page can be sent. Nothing is sent until you press Calculate, and pressing it again on an unchanged drawing sends nothing.

Errors and warnings

A refused edit is explained on the drawing itself and leaves the drawing as it was. If the calculation service does not answer, the page says so, keeps your drawing and the last result, and lets you press Calculate again. A NOT VALID: status means the rows could not be measured and no figures are shown.

Sources and further reading

• Bart Braden, "The Surveyor's Area Formula," The College Mathematics Journal 17, no. 4 (1986): 326-337. doi:10.1080/07468342.1986.11972974. The derivation of the area of a polygon from the coordinates of its corners, the relationship this calculator's spreadsheet evaluates.
• Eric W. Weisstein, "Polygon Area," MathWorld, Wolfram Research. mathworld.wolfram.com/PolygonArea.html. The same formula in summation form, with the conditions it needs: a flat polygon whose sides do not cross.
• National Institute of Standards and Technology, Guide for the Use of the International System of Units (SI), NIST Special Publication 811, Appendix B.8. nist.gov. The exact factors behind the US/Metric switch: 1 inch is 2.54 cm, 1 foot is 0.3048 m, and 1 square foot is 0.09290304 m².

These sources describe the method; they do not certify a particular drawing or its result. The area is only as good as the drawing: measure the space you are drawing, and check the figures before relying on them for an order, a quote or a contract.