How this instrument works
The Shoelace Formula finds a polygon's area directly from its vertices' coordinates, with no need to split the shape into triangles or rectangles first. For a four-sided figure with corners (x₁,y₁) through (x₄,y₄), listed IN ORDER walking around the boundary (either all clockwise or all counterclockwise), the area is A = ½|x₁y₂−x₂y₁+x₂y₃−x₃y₂+x₃y₄−x₄y₃+x₄y₁−x₁y₄|.
The formula gets its name from the crisscrossing pattern the paired multiplications trace when written out by hand, resembling a shoe's laces threading through its eyelets. Each term multiplies one vertex's x-coordinate by the NEXT vertex's y-coordinate, and subtracts the reverse pairing — a pattern that, when summed all the way around the polygon and taken as half its absolute value, gives the exact enclosed area regardless of how irregular or skewed the shape is.
The one requirement that matters is listing the vertices in a consistent order around the boundary — if they're entered out of sequence (crossing back and forth rather than walking steadily around the perimeter), the formula can return an incorrect or even negative-looking intermediate result, since it implicitly assumes a simple, non-self-intersecting polygon traced in one consistent direction.
- Enter the first corner's coordinates into the Vertex 1: x and Vertex 1: y fields.
- Enter the remaining three corners, in order walking around the shape's boundary, into the Vertex 2, 3, and 4 fields.
- Read Area: the sheet applies the Shoelace Formula to the four vertices.
Worked example — an irregular quadrilateral
A four-sided figure has corners at (0,0), (4,0), (5,3), and (1,4), listed in order around the boundary. Applying the Shoelace Formula gives an area of exactly 14.5 — reached directly from the coordinates alone, with no need to manually split the irregular shape into a triangle and a rectangle or any other combination first.
As a sanity check, a plain rectangle with corners (0,0), (4,0), (4,3), (0,3) gives an area of exactly 12 through this same formula — matching the familiar length-times-width result exactly, confirming the Shoelace Formula correctly reduces to simpler methods for regular shapes too.
Questions
What is the Shoelace Formula?
A method for finding a polygon's area directly from its vertices' coordinates, without needing to split the shape into triangles or rectangles by hand. It works for any simple (non-self-intersecting) polygon, however irregular its shape.
Why does the order of the vertices matter?
The formula assumes the vertices are listed walking consistently around the polygon's boundary, either all clockwise or all counterclockwise. Entering them out of that order can produce an incorrect result, since the formula implicitly traces the shape's perimeter in the order given.
Why is there an absolute value in the formula?
The raw sum before taking the absolute value can come out negative depending on whether the vertices were listed clockwise or counterclockwise — area itself is always non-negative, so the absolute value strips away that direction-dependent sign.
Does this work for triangles or shapes with more than four sides?
Yes — the Shoelace Formula generalizes to any number of vertices; this page is scoped specifically to four-sided figures (quadrilaterals), a very common practical case, but the identical crisscross pattern extends to any polygon with more or fewer corners.
What if the four points don't form a simple (non-crossing) shape?
The formula assumes the boundary doesn't cross itself — for a self-intersecting arrangement of the same four points, the result may not represent the visually enclosed area in the way expected, since the underlying assumption of a simple polygon traced in one direction no longer holds.