Area from coordinates
When a shape is given by the coordinates of its corners, such as a plot of land from a survey, a room from a floor plan or a polygon in a drawing program, there is no need to split it into triangles. The shoelace formula gives the area directly from the list of vertices.
The shoelace formula
For vertices (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ) listed in order around the shape:
Area = ½ × |x₁y₂ − x₂y₁ + x₂y₃ − x₃y₂ + … + xₙy₁ − x₁yₙ|
Write the coordinates in two columns, repeat the first point at the bottom, and multiply diagonally in both directions: the crisscross pattern looks like the laces of a shoe. The formula is also known as Gauss’s area formula or the surveyor’s formula.
Without the absolute value, the sum gives a signed area: positive when the vertices run counterclockwise, negative when they run clockwise.
Perimeter and centroid
The perimeter is the sum of the distances between consecutive vertices, including the closing edge from the last vertex back to the first.
The centroid, the balance point of a flat plate with that shape, is:
Cx = Σ (xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) ÷ (6A) and similarly for Cy, using the signed area A.
Worked example
A pentagon with vertices (0, 0), (4, 0), (5, 3), (2, 5) and (−1, 3):
- Cross products: 0, 12, 19, 11 and 0, for a sum of 42.
- Area: 42 ÷ 2 = 21 square units.
- The sum is positive, so the vertices run counterclockwise.
- Perimeter: 4 + 3.1623 + 3.6056 + 3.6056 + 3.1623 = 17.5357.
- Centroid: (2.0000, 2.1905).
Rules for correct results
- List the vertices in order around the boundary. Jumping across the shape creates crossing edges and a wrong area.
- The polygon must be simple: its edges must not cross. Concave shapes, such as an L or an arrow, are fine.
- Do not repeat the first vertex at the end; the calculator closes the shape for you.
- The area is in the square of the coordinate unit: square feet if the coordinates are in feet.
Uses
Surveyors compute parcel areas from corner coordinates. GIS software applies the same formula to map polygons, on projected coordinates rather than latitude and longitude. Game developers use the sign of the area to tell whether a polygon’s vertices are wound clockwise. For a regular polygon defined by its side length, the regular polygon calculator is quicker.