SamCalculator

Polygon Area Calculator

Calculate the area of any closed polygon from its vertex coordinates using the Shoelace (Surveyor's) formula.

Choose a shape

Polygon inputs

A = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|

Enter the vertices of your polygon in order — clockwise or counter-clockwise.

v1
v2
v3
v4

Master area formula library

Rectangle

A = l × w

Length times width.

Square

A = s²

A rectangle with equal sides.

Triangle

A = ½ × b × h

Half the base times the height.

Triangle (Heron)

A = √[s(s−a)(s−b)(s−c)]

From three side lengths only.

Trapezoid

A = ½(b₁ + b₂) × h

Average of the parallel bases × height.

Circle

A = π × r²

Pi times the radius squared.

Sector

A = ½ × r² × θ

Radians for θ.

Ellipse

A = π × a × b

Semi-major × semi-minor × π.

Parallelogram

A = b × h

Base times perpendicular height.

Rhombus

A = ½ × d₁ × d₂

Half the product of the diagonals.

Kite

A = ½ × d₁ × d₂

Same as the rhombus diagonal rule.

Regular polygon

A = ½ × n × s × a

n sides of length s, apothem a.

Polygon

A = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|

Shoelace / Surveyor's formula.

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What Is the Polygon Area Formula?

Polygon area calculations let you find the two-dimensional extent of a polygon from its defining measurements. The formula A = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| captures the relationship between the inputs and the area; the calculator applies it instantly, returns a full unit-conversion panel, and shows every substitution step.

The formula used on this page is A = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|. The calculator applies it instantly, returns full unit conversions, and shows every substitution in the step-by-step panel.

How the Calculation Works

The polygon area formula A = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| comes from classical Euclidean geometry. The inputs you enter are substituted directly into the formula; the result is the area in the square of your input unit. The step-by-step panel below the result mirrors how the same calculation would be written on paper.

6 Ways to Use This Polygon Calculator

1

Geometry coursework

Solve textbook polygon area problems and see the full step-by-step working alongside the final answer.

2

Practical sizing

Estimate floor area, surface area, or footprint of any polygon-shaped object — from a tile to a parcel of land.

3

Unit conversions

Enter dimensions in any common unit and instantly read the area in mm², cm², m², ha, km², in², ft², yd², acres, and mi².

4

Engineering checks

Quickly verify polygon area calculations done by hand, in a spreadsheet, or in CAD software.

5

Education & teaching

Use the diagram, step-by-step panel, and area conversion panel as a teaching aid for geometry classes.

6

Reference & quick checks

Look up the polygon formula, the result format, and equivalent unit values in a single page.

Best Practices

  • Use consistent units. Every length input must use the same unit. The output area is automatically in the square of that unit.
  • Choose the correct measurement. Read the input labels carefully — many polygon formulas distinguish between perpendicular height and slant length.
  • Don't round inputs prematurely. Keep the full precision of your inputs; the calculator only rounds the displayed result.

Why Polygon Area Matters

Architecture & construction

Polygon shapes appear in floor plans, panel layouts, roof sections, and structural members — accurate area drives material take-offs.

Engineering & design

Cross-sections, plates, and panels are often polygon-shaped; their area drives stress, weight, and cost calculations.

Surveying & land use

Surveyors decompose irregular plots into polygon-shaped patches; summing the areas gives the total parcel size.

Academia & education

Polygon area is a standard topic in primary, secondary, and undergraduate geometry curricula worldwide.

Common Mistakes

  • Mixing length and width units. If one side is in feet and the other in inches, convert first or the area will be off by a factor of 12.
  • Forgetting to square the unit. Area is reported in unit² (m², ft², etc.). The calculator adds the square automatically.
  • Confusing perimeter and area. Perimeter is a linear measurement around the boundary; area is the 2D space inside it.

Core Formula

Formula

A = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|

Area is always returned in your input unit squared. The conversion panel translates the result into mm², cm², m², ha, km², in², ft², yd², acres, and mi² so you can compare across unit systems.

Methodology & Accuracy

Every calculation runs entirely in your browser using JavaScript's full double-precision arithmetic (about 15–17 significant digits internally). The displayed result is rounded to four significant figures by default; numbers outside the 0.0001 to 999,999,999 range switch automatically to scientific notation. The polygon formula used here is the standard Euclidean-geometry identity, sourced from the same textbooks used in secondary-school and university geometry courses.

Frequently Asked Questions

The Shoelace (or Surveyor's) formula computes the area of any simple polygon from its vertex coordinates: A = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|, where the sum wraps around the polygon. The name comes from the cross-multiplying pattern, which resembles tying a shoelace.

The vertices must be listed in order around the polygon — either all clockwise or all counter-clockwise. The calculator reports which direction you used. The result is the same regardless of direction because the formula takes the absolute value.

Yes, as long as the polygon is simple (the edges don't cross each other). The Shoelace formula handles convex and concave polygons identically.

It uses the standard centroid formulas for a polygon: Cx = (1 / 6A) × Σ(xᵢ + xᵢ₊₁)(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ), with an analogous formula for Cy. These weight each edge by its signed contribution to the area.

The calculator supports up to 12 vertices for ease of input. For larger polygons (such as GIS outlines), use a GIS tool — the underlying Shoelace formula scales to any number of points.

Related Calculators

Other geometry and measurement tools that pair with area calculations.