About the Polygon Area from Coordinates
This calculator finds the area of any simple polygon — triangle, quadrilateral, an irregular plot of land, a room outline — from the coordinates of its corners, using the shoelace (surveyor’s or Gauss) formula. Type or paste one “x, y” pair per line, in order around the shape, and it returns the area, the perimeter, the centroid (centre of area) and whether you listed the points clockwise or counter-clockwise.
It is handy for coordinate-geometry homework, for surveyors and GIS users working with projected coordinates in metres or feet, for estimating the area of an irregular lot from a site plan, and for game and CAD developers checking polygon winding.
The vertices must be listed in order along the boundary, and the polygon must not cross itself; a self-intersecting “bow-tie” shape gives a misleading net area. Latitude/longitude pairs must be projected to a flat grid first, because degrees are not equal-length units.
With the default inputs, the area is 21. Change any value above to recalculate instantly.
How to use the polygon area from coordinates
- 1List the polygon’s corners in order around the boundary (either direction).
- 2Enter one “x, y” pair per line — no need to repeat the first point.
- 3Read the area, perimeter and centroid.
- 4Check the cross-product table to follow the shoelace steps.
Formula and method
The shoelace formula multiplies each vertex’s x by the next vertex’s y, subtracts the next x times the current y, adds these cross-products all the way round (wrapping from the last vertex back to the first) and halves the absolute total. Written in two columns, the criss-cross multiplication looks like lacing a shoe, hence the name.
Each cross-product is twice the signed area of the triangle formed by the origin and one edge; summing them cancels everything outside the polygon. The sign tells you the winding: positive for counter-clockwise, negative for clockwise. The centroid uses the same cross-products as weights, and the perimeter adds the straight-line distance of each edge.
- (xᵢ, yᵢ)
- Coordinates of vertex i, in boundary order
- n
- Number of vertices (vertex n + 1 wraps to vertex 1)
- A
- Signed area (absolute value is the area)
- Cx, Cy
- Centroid coordinates
Worked examples
Irregular pentagon
The cross-products for (1,1), (5,1), (6,4), (3,6), (0,4) are −4, 14, 24, 12 and −4, which sum to 42. Half of that gives an area of 21 square units; the positive sign means the points were listed counter-clockwise.
3-4-5 right triangle
The only non-zero cross-product is 4 × 3 = 12, so the area is 6 — matching ½ × 4 × 3. The perimeter is 4 + 5 + 3 = 12 and the centroid is the average of the corners, (4/3, 1).
L-shaped room outline
An L-shape made from a 6 × 2 strip plus a 2 × 3 leg has area 12 + 6 = 18, which the shoelace sum confirms without splitting the shape into rectangles. Its outline is 22 units long.
Frequently asked questions
What is the shoelace formula?+
It computes a polygon’s area from its vertex coordinates: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. It works for any simple polygon — convex or concave — as long as the vertices are listed in order around the edge.
Does the order of the points matter?+
Yes. Points must follow the boundary, clockwise or counter-clockwise. Direction only changes the sign of the signed area, but jumping across the shape creates a self-intersecting outline and a wrong area.
Can I use GPS latitude and longitude?+
Not directly. Degrees of longitude shrink toward the poles, so first convert coordinates to a flat projection such as UTM in metres; then the shoelace result is in square metres.
Why is my area zero or too small?+
A zero area means the points are collinear or cancel out. A too-small area usually means the outline crosses itself because points were entered out of order; the crossing loops subtract from each other.
Do I need to repeat the first point at the end?+
No. The formula wraps from the last vertex back to the first automatically. If you do repeat it, the duplicate closing point is ignored.