The Area from Coordinates Calculator calculates the area of a polygon from the x, y coordinates of its corners using the shoelace formula.
It cross-multiplies neighboring coordinates, adds the results and halves the absolute total: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. The calculator also gives the perimeter and the centroid (center of area), and draws the shape so you can check the corner order.
Area from Coordinates Calculator
Area, perimeter and centroid of any polygon from the x, y coordinates of its corners, using the shoelace formula — with a graph.
Formula
Area (shoelace formula):
A = ½ |Σ (xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ)|
Centroid:
Cx = Σ (xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) ÷ 6A
Cy = Σ (yᵢ + yᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) ÷ 6A
Where (xᵢ, yᵢ) are the corners listed in order, and the last corner wraps around to the first. (For the centroid, A is the signed area.)
How the Calculation Works
Step 1: List the corners in order around the shape, clockwise or counterclockwise.
Step 2: For each pair of neighbors, multiply diagonally: x of the first × y of the second, minus x of the second × y of the first. Writing the coordinates in two columns makes these crossing lines look like laced shoes.
Step 3: Add all the products, take the absolute value and divide by 2.
Variables Explained
| Input / output | Meaning |
|---|---|
| x, y | Position of each corner |
| Area | Space inside the polygon |
| Perimeter | Sum of the side lengths |
| Centroid (Cx, Cy) | Balance point of the shape’s area |
Example Calculation
Given: (1, 1), (9, 2), (11, 7), (5, 10), (0, 6) in cm
| Pair | xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ |
|---|---|
| (1, 1) → (9, 2) | 1×2 − 9×1 = −7 |
| (9, 2) → (11, 7) | 9×7 − 11×2 = 41 |
| (11, 7) → (5, 10) | 11×10 − 5×7 = 75 |
| (5, 10) → (0, 6) | 5×6 − 0×10 = 30 |
| (0, 6) → (1, 1) | 0×1 − 1×6 = −6 |
Sum = 133
Result: A = 133 ÷ 2 = 66.5 cm²; perimeter ≈ 31.66 cm; centroid ≈ (5.25, 5.17)
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10,000 cm² = 10.7639 ft² |
| 1 ft² | 144 in² |
| 1 acre | 43,560 ft² |
Shape Properties
The shoelace formula works for any simple polygon — convex or with inward corners — as long as the sides don’t cross. The sign of the sum tells you the direction: positive for counterclockwise, negative for clockwise.
Dimensions Explained
Getting coordinates
Coordinates can come from graph paper, a CAD drawing, a site plan or measurements from a fixed starting corner. All coordinates must use the same unit.
Frequently Asked Questions
How do you find the area of a polygon from coordinates?
Use the shoelace formula: cross-multiply neighboring corners, add, take the absolute value and halve.
Does the order of the points matter?
Yes. They must go around the shape in order. Direction (clockwise or counterclockwise) doesn’t change the area.
How do you find the area of a triangle from coordinates?
A = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|, which is the shoelace formula for three points.
Why is it called the shoelace formula?
The diagonal cross-multiplications look like the crisscross lacing of a shoe.
Can I use negative coordinates?
Yes. Negative values work normally.
