Math
Irregular Polygon Area Calculator
Paste the vertices of a polygon, one per line and in order around the boundary, and mathcheck applies the shoelace (surveyor's) formula with each xi·yi+1 - xi+1·yi term written out.
The area is exact for decimal coordinates, and you also get the perimeter from the side lengths, the centroid, whether the vertices run clockwise or counterclockwise, a warning if any sides cross, and a drawing of the polygon with its centroid marked.
Type x, y pairs in order around the boundary (clockwise or counterclockwise). Do not repeat the first vertex at the end; the calculator closes the shape.
Try an example
Result
Area
—
- Perimeter
- —
- Centroid
- —
- Vertex order
- —
- Number of vertices
- —
- Shape check
- —
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Study path
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The shoelace formula
List the The turning point of a parabola, or a corner point of a shape. in order around the polygon and write the coordinates in two columns. Multiply diagonally down-right, multiply diagonally down-left, subtract, and halve the How far a number sits from zero, so the answer is never negative.; the criss-cross pattern of the multiplications looks like laced shoes, which gives the formula its name. It works for any simple polygon, convex or concave, and is exact for decimal coordinates.
Pentagon (0,0), (6,0), (7,3), (3,6), (-1,3)
Terms: 0·0 - 6·0 = 0; 6·3 - 7·0 = 18; 7·6 - 3·3 = 33; 3·3 - (-1)·6 = 15; (-1)·0 - 0·3 = 0. Sum = 66, so A = 33. Positive sum means the vertices run counterclockwise.
| Sign of the sum | Vertex order | Area |
|---|---|---|
| positive | counterclockwise | sum / 2 |
| negative | clockwise | |sum| / 2 |
| zero | collinear or self-cancelling | 0 |
How to use this calculator
- Type each vertex as x, y on its own line, following the boundary either clockwise or counterclockwise.
- Do not repeat the first vertex at the end; if you do, the calculator ignores the duplicate.
- Read the exact area in the header, then the perimeter, centroid, orientation, and shape check.
- Look at the plot to confirm the shape is the one you meant; crossing sides mean the order is wrong.
- Open Show the work to see every shoelace term, the side lengths, and the centroid sums.
How to read the answer
The area is exact whenever the coordinates are exact decimals. A positive shoelace sum means counterclockwise order and a negative one clockwise; the area is the same either way. The centroid is the balance point of the region (not the average of the vertices, unless the polygon is a triangle). A self-intersecting warning means the number shown is a signed net area, not the area of the visible figure.
Common mistakes and edge cases
- Listing vertices out of boundary order, which creates crossing sides and a wrong (usually smaller) area.
- Forgetting the last term that wraps from the final vertex back to the first.
- Skipping the absolute value and reporting a negative area for clockwise input.
- Halving twice, or not at all. The sum of the cross terms is twice the area.
- Averaging the vertices to get the centroid. That gives the vertex centroid, which differs from the area centroid for anything but a triangle.
Worked examples
Pentagon
(0,0), (6,0), (7,3), (3,6), (-1,3)
Area
33
Rectangle, clockwise
(0,0), (0,3), (4,3), (4,0)
Area
12
Concave L-shape
(0,0), (4,0), (4,1), (1,1), (1,4), (0,4)
Area
7
Decimal coordinates stay exact
Triangle (0,0), (0.1,0), (0.1,0.3)
Area
3/200 (0.015)
Negative coordinates, semicolon separated
(-2,-1); (2,-1); (2,2); (-2,2)
Area
12
Bow-tie (self-intersecting)
(0,0), (2,2), (2,0), (0,2)
Area
0
Collinear points
Area
0
Too few vertices
Area
—
Malformed line
Area
—
Frequently asked questions
How do I find the area of an irregular polygon from coordinates?+
Use the shoelace formula: list the vertices in order, sum xi·yi+1 - xi+1·yi around the shape (including the wrap from the last vertex to the first), and take half the absolute value.
Does the vertex order matter?+
The vertices must follow the boundary, but clockwise or counterclockwise both work; the sign of the sum flips and the absolute value removes it. Scrambled order makes sides cross and gives a wrong area.
Does the shoelace formula work for concave polygons?+
Yes, as long as the boundary does not cross itself. Concave notches are handled automatically because the cross terms for the notch subtract the right amount.
What does the calculator do with a self-intersecting polygon?+
It still computes the shoelace sum but warns you. For a bow-tie the two lobes have opposite orientation and cancel, so the formula returns 0 even though the figure clearly has area.
How is the centroid computed?+
With weighted sums of consecutive vertex pairs divided by six times the signed area. It is the center of mass of the filled region, which differs from the plain average of the vertices unless the polygon is a triangle.
Can I use this for survey plots or land parcels?+
Yes; the shoelace formula is also called the surveyor's formula. Enter the corner coordinates in traverse order and the area comes out in the square of whatever unit the coordinates use.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
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Last updated: September 4, 2026