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Shoelace formula calculator

Compute a polygon's area with the shoelace formula (Gauss's area formula): sum the cross-products of consecutive vertex coordinates, halve, and take the absolute value — A = ½·|Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|. It also gives the signed area, whose sign tells you the winding direction, plus the perimeter and centroid.

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Open Shoelace formula calculator and you get an answer straight away, with no account to create. It covers sum the cross-products of consecutive vertex coordinates, halve, and take the absolute value — A = ½·|Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|. It also gives the signed area, whose sign tells you the winding direction, plus the perimeter and centroid — adjust any of them and the result follows immediately.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is Shoelace formula calculator?

Compute a polygon's area with the shoelace formula (Gauss's area formula): sum the cross-products of consecutive vertex coordinates, halve, and take the absolute value — A = ½·|Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|. It also gives the signed area, whose sign tells you the winding direction, plus the perimeter and centroid.

What does a concrete case look like?

(0,0), (4,0), (0,3) → A = 6 — the tool shows every step in between, not just the final figure.

What does it take into account?

It factors in sum the cross-products of consecutive vertex coordinates, halve, and take the absolute value — A = ½·|Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|. It also gives the signed area, whose sign tells you the winding direction, plus the perimeter and centroid. Change any of them and the output follows immediately.

When would I actually use this?

Homework and site work alike: how much paint a wall takes, how much water a tank holds, whether a corner is square — and, on the coordinate side, the slope between two points, the distance across a plan, the midpoint of a span.

What is the most common mistake?

Mixing units inside one shape. Entering a length in centimetres and a width in metres gives an area wrong by a factor of a hundred, and the result looks perfectly plausible.

How is Shoelace formula calculator different from Distance Formula Calculator?

They sit next to each other but answer different questions: Distance Formula Calculator is the one to open when you need it to find the distance between two points on an interactive coordinate plane, with the Pythagorean working, the midpoint, the slope and the direction angle. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Triangle area calculator (Heron's formula) is the closest one after this: Compute a triangle's area from the length of its three sides.

Where do the figures come from?

The formulas are the classical ones and the results are exact to the precision shown — geometry does not change with the year or the country, so nothing here needs updating.

Further reading

All guides →
How-toHow to Find the Distance Between Two PointsUse the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) to measure the straight-line distance between two points on a plane, with a worked example and the midpoint too.How-toHow to Calculate the Slope of a Line: Rise Over RunFind the slope of a line from two points with m = (y₂ − y₁) / (x₂ − x₁), and read what a positive, negative, zero, or undefined slope tells you.How-toHow to Use the Quadratic Formula: Solving ax² + bx + c = 0Solve any quadratic equation with x = (−b ± √(b² − 4ac)) / 2a, and read the discriminant to know how many real solutions to expect.How-toHow to Find the Area of a Triangle: Base × Height and Heron's FormulaTwo reliable ways to find a triangle's area: half base times height when you know the height, and Heron's formula when you only know the three sides.ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.