Irregular Shape Area Calculator

An irregular shape has no single formula of its own, because it has no fixed shape to base one on. What it does have is corners, and the positions of the corners are enough. This method is known as the shoelace formula, after the criss-cross pattern its multiplications make on paper. It handles an outline with any number of straight sides, including ones with inward-pointing corners.

Formula

A = ½ × |Σ (xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ)|

A

the area enclosed by the outline

xᵢ, yᵢ

the position of each corner, taken in order around the edge

xᵢ₊₁, yᵢ₊₁

the next corner along, with the last corner wrapping back to the first

Getting the coordinates

Pick any convenient corner as 0, 0. Choose one direction as x and a right angle to it as y, then measure how far along and how far across each corner sits. On a drawing, grid paper makes this quick. On the ground, stretch a long tape as a baseline and measure straight out from it to each corner.

Enter the corners in the order you would meet them walking around the edge. Direction does not matter, and neither does the starting point. Latitude and longitude will not work directly, because degrees are not lengths, so convert those to distances from a start point first.

A worked example

Take an L-shaped room with corners at (0, 0), (12, 0), (12, 5), (5, 5), (5, 9) and (0, 9), in feet. For each corner, multiply its x by the next corner's y and subtract the next corner's x times its own y. The six results add up to 160, and half of that gives an area of 80 ft².

Composite figures: split into rectangles

Plenty of shapes break neatly into rectangles, and then coordinates are more work than they are worth. The same L-shaped room is a 12 by 5 rectangle plus a 5 by 4 rectangle, so 60 + 20 = 80 ft². Or treat it as a full 12 by 9 rectangle with a 7 by 4 corner missing. Enter that second piece with a minus sign, as -7, 4, and you get 108 − 28 = 80 ft², the same answer.

Keep the pieces from overlapping, since any overlap is counted twice. For composite figures with triangles or circles, work out those parts with their own calculators and add them to the result.

Outlines that cross themselves

The shoelace formula assumes the edge never crosses itself. If corners are entered out of order, the outline becomes a figure-eight, and the two loops partly cancel, which produces a number that looks plausible but is wrong. The calculator checks for this and reports an error instead. It reports the same if an edge touches another or doubles back on itself. The fix is almost always to reorder the lines.

Curved edges

Coordinates connect corners with straight lines, so a curve has to be approximated with several short sides. The more points you use, the closer you get. A circle of radius 10 has an area of about 314.16. Twelve evenly spaced points around it give an area of 300, about 4.5% low, while 24 points give about 310.58, about 1.1% low. An outline drawn with its points on the curve always loses a little on a bulging edge.

Frequently asked questions