Land surveys, floor plans, game maps and GPS boundaries all describe shapes as lists of corner coordinates. The shoelace formula turns such a list directly into an area — no need to split the shape into triangles or rectangles. This calculator applies it to any polygon with up to 100 vertices, shows every cross-product term, and also reports the perimeter, the centroid, the direction the vertices run, and whether the shape is convex.
How to use the polygon area calculator
- Enter the vertices in order around the shape, one “x, y” pair per line. Clockwise or counterclockwise both work; don’t repeat the first point at the end.
- Choose the unit of the coordinates.
- Read the area on the tape. The table lists each cross term, and the diagram plots the polygon with numbered vertices and its centroid.
The shoelace formula
Each term is twice the signed area of the triangle formed by the origin and one edge. Edges running counterclockwise add area and edges running back subtract it, so the total is the area enclosed — no matter where the origin is.
The centroid (center of mass of a uniform plate) uses the same terms:
Worked example
Find the area of the pentagon with vertices (1, 1), (7, 2), (8, 6), (4, 9) and (0, 5).
| Edge | xi·yi+1 | xi+1·yi | Difference |
|---|---|---|---|
| 1 → 2 | 1 × 2 = 2 | 7 × 1 = 7 | −5 |
| 2 → 3 | 7 × 6 = 42 | 8 × 2 = 16 | 26 |
| 3 → 4 | 8 × 9 = 72 | 4 × 6 = 24 | 48 |
| 4 → 5 | 4 × 5 = 20 | 0 × 9 = 0 | 20 |
| 5 → 1 | 0 × 1 = 0 | 1 × 5 = 5 | −5 |
Sum: −5 + 26 + 48 + 20 − 5 = 84.
Area: ½ × |84| = 42 square units. The sum is positive, so the vertices run counterclockwise.
Also: the perimeter is about 24.99 units, the centroid is at about (3.97, 4.62), and the shape is convex.
Pick's theorem: the boundary passes through B = 8 grid points, so there are I = 42 − 8/2 + 1 = 39 grid points inside.
Practical tips
- Close the loop mentally, not in the data. The formula automatically connects the last vertex back to the first.
- Keep the order consistent. Jumping across the shape creates crossing edges and a meaningless result.
- Use a local origin for big numbers. With map coordinates in the millions, subtract a nearby reference point from every vertex first to avoid rounding loss; the area does not change.
- Convert units once. Coordinates in feet give square feet; divide by 43,560 for acres.
- Latitude and longitude are not flat coordinates. Project them onto a flat map grid (such as UTM or a state plane system) before using the formula.
Convex or concave?
A polygon is convex when every interior angle is less than 180°, which the calculator detects by checking that consecutive edges always turn the same way. Concave shapes — an L-shaped room, a star — are handled just as well by the shoelace formula, but some properties change: the centroid of a concave shape can lie outside it.
Related tools
For a triangle from three side lengths or a base and height, use the triangle area calculator. Edge lengths come from the distance calculator, edge midpoints from the midpoint calculator, and edge slopes from the slope calculator. For polygons with equal sides and angles, the regular polygon calculator is quicker.
Frequently asked questions
What is the shoelace formula?
For vertices (x₁, y₁) … (xₙ, yₙ) listed in order, area = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|, wrapping from the last vertex back to the first. The criss-cross pattern of the products looks like lacing a shoe, hence the name.
Does the order of the vertices matter?
Yes. List them in order around the boundary, either clockwise or counterclockwise. Counterclockwise gives a positive signed area and clockwise a negative one; the calculator reports the absolute value as the area.
Does it work for concave polygons?
Yes, for any simple polygon — one whose edges do not cross. If edges cross, as in a bow-tie shape, the formula returns a net signed area in which overlapping loops can cancel, and the calculator warns you.
What is Pick's theorem?
For a polygon whose vertices all have whole-number coordinates, area = I + B/2 − 1, where I counts grid points strictly inside and B counts grid points on the boundary. The calculator uses it to report those counts.