Polygon Area Calculator
Calculate the area of any polygon — 3 to 8+ sides, irregular lots, uneven sides, or coordinates. Convert results to acres, hectares, square feet & more.
A polygon area calculator computes the area of any multi-sided land parcel — from triangles (3 sides) to octagons (8 sides) and beyond — and converts the result to acres, hectares, square feet (sq ft), square meters (m²), and other units. This free online tool supports two input methods: enter individual side lengths for regular or irregular polygons, or enter vertex coordinates for exact area calculation using the Shoelace formula (Gauss's area formula).
The polygon area calculator serves property owners measuring irregular lots with uneven sides, land surveyors computing parcel areas from GPS coordinates, real estate professionals estimating acreage for non-rectangular properties, and construction teams calculating site coverage for zoning compliance. This page covers what a polygon is, how to calculate polygon area by side lengths and coordinates, formulas for 3-sided through 8-sided shapes, the Shoelace formula explained step by step, and answers to 10 frequently asked questions about polygon land measurement.
What is a Polygon?
A polygon is a closed 2D shape with 3 or more straight sides. In land measurement, almost every property boundary forms a polygon — from simple triangular plots to complex parcels with 6, 8, or more sides. Polygons are classified by their number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), and octagon (8).
Polygons are further divided into regular (all sides and angles equal) and irregular (sides and/or angles differ). Most real-world land parcels are irregular polygons — properties with uneven sides, different lengths, and non-right angles. The polygon area calculator above handles both regular and irregular polygons with 3 to 8+ sides.
The green overlay shows 1 acre relative to the selected field. Toggle between comparisons to see how an acre measures up.
Polygon Types & Interior Angles
The interior angles of a regular polygon sum to (n − 2) × 180°, where n is the number of sides. Each interior angle of a regular polygon equals (n − 2) × 180° ÷ n. The following table shows properties for regular polygons with 3 to 8 sides:
| Polygon | Sides | Interior Angle | Area Formula (side = s) |
|---|---|---|---|
| Triangle | 3 | 60° | (√3 / 4) × s² |
| Square | 4 | 90° | s² |
| Pentagon | 5 | 108° | (√(25 + 10√5) / 4) × s² |
| Hexagon | 6 | 120° | (3√3 / 2) × s² |
| Heptagon | 7 | ≈128.57° | ≈ 3.6339 × s² |
| Octagon | 8 | 135° | 2(1 + √2) × s² |
An acre has no fixed shape — all these shapes have exactly the same area: 43,560 sq ft.
Lot Size Reference: What Does an Acre Look Like?
One acre covers about 75% of an American football field (including end zones) or 89% of a standard FIFA soccer field. Here are 5 real-world size comparisons for one acre of land:
Land parcels in real estate and agriculture are sold in fractional portions of an acre. Here is a quick reference for common lot sizes in square feet:
Click a fraction to see how much of an acre it covers. The green area shows the selected portion.
How to Calculate Polygon Area
To calculate the area of a polygon, choose the method based on the information available: use the regular polygon formula for shapes with equal sides, Heron's formula for triangles with 3 known sides, or the Shoelace formula for any polygon with known vertex coordinates. The polygon area calculator above performs all three methods automatically.
Method 1: Regular Polygon Formula
For a regular polygon where all sides are equal length, use the following formula:
Example: A regular hexagon with 100 ft sides:
(6 × 100²) ÷ (4 × tan(π/6)) = 25,980.76 sq ft = 0.5966 acres
Method 2: Heron's Formula (Triangles — 3 Sides)
For a triangle where all 3 side lengths are known, use Heron's formula to calculate the area without needing the height:
Example: A triangle with sides 300 ft, 400 ft, and 500 ft:
s = (300 + 400 + 500) ÷ 2 = 600
Area = √(600 × 300 × 200 × 100) = 60,000 sq ft = 1.378 acres
Method 3: Brahmagupta's Formula (Quadrilaterals — 4 Sides)
For a quadrilateral with 4 known sides, Brahmagupta's formula gives the maximum possible area (achieved when the quadrilateral is cyclic — inscribed in a circle):
Example: A quadrilateral with sides 200 ft, 250 ft, 300 ft, 350 ft:
s = (200 + 250 + 300 + 350) ÷ 2 = 550
Area = √(350 × 300 × 250 × 200) = 72,457 sq ft ≈ 1.664 acres
Method 4: Shoelace Formula (Any Polygon — Coordinates)
The Shoelace formula (also called the Surveyor's formula or Gauss's area formula) calculates the exact area of any polygon when you know the (x, y) coordinates of each vertex:
P1 = (0, 0), P2 = (200, 0), P3 = (250, 180), P4 = (30, 200)
= ½ × |(0×0 − 200×0) + (200×180 − 250×0) + (250×200 − 30×180) + (30×0 − 0×200)|
= ½ × |0 + 36,000 + 44,600 + 0| = 40,300 sq ft = 0.925 acres
5 steps to calculate polygon area using the Shoelace formula:
- Identify all corners (vertices) of the polygon and list them in sequential order — either clockwise or counterclockwise
- Record each vertex as an (x, y) coordinate in consistent units (feet, meters, etc.)
- For each pair of consecutive vertices, compute xᵢ × yᵢ₊₁ and xᵢ₊₁ × yᵢ
- Sum all the cross products, take the absolute value, and divide by 2 to get the area in square units
- Divide the square feet total by 43,560 to convert to acres
Measuring Polygon Area with Google Maps
Google Maps can measure the area of any polygon-shaped land parcel using the built-in distance measurement tool. Here are 5 steps to measure polygon acreage with Google Maps:
- Open Google Maps (maps.google.com) and search for the property address or GPS coordinates
- Switch to Satellite view by clicking the "Layers" button in the bottom-left corner
- Right-click on a starting corner of the property boundary and select "Measure distance"
- Click on each corner of the polygon boundary to outline all sides of the land parcel
- Close the shape by clicking on the starting point — Google Maps displays the total area in square feet (sq ft) and square meters (m²). Divide by 43,560 for acres
Measuring Polygon Area with Google Earth
Google Earth Pro provides more precise polygon area measurements than Google Maps with a dedicated polygon measurement tool. Here are 6 steps:
- Download and install Google Earth Pro (free) from earth.google.com
- Search for the property address or coordinates in the search bar
- Select the "Add Polygon" tool from the toolbar (or go to Add > Polygon)
- Click on each corner of the property boundary to trace all sides of the polygon
- Close the polygon by clicking on the starting point
- View the calculated area in the "Measurements" tab — Google Earth Pro displays results in acres, hectares, square feet (sq ft), and square meters (m²)
Unit Conversion Reference Table
The following reference table provides area unit conversions between 10 common land measurement units used in real estate, agriculture, and surveying:
| From | To | Multiply By |
|---|---|---|
| Acres | Square Feet | 43,560 |
| Acres | Square Meters | 4,046.86 |
| Acres | Hectares | 0.404686 |
| Acres | Square Miles | 0.0015625 |
| Acres | Square Yards | 4,840 |
| Acres | Square Kilometers | 0.00404686 |
| Hectares | Acres | 2.471054 |
| Square Feet | Acres | 0.00002296 |
| Square Meters | Acres | 0.000247105 |
| Square Miles | Acres | 640 |
Polygon Area in Different Units
Use the following quick reference when converting polygon area calculations between common units:
| Acres | Square Feet | Square Yards | Square Meters |
|---|---|---|---|
| 0.25 acres | 10,890 ft² | 1,210 yd² | 1,011.71 m² |
| 0.33 acres | 14,375 ft² | 1,597 yd² | 1,335.46 m² |
| 0.5 acres | 21,780 ft² | 2,420 yd² | 2,023.43 m² |
| 1 acre | 43,560 ft² | 4,840 yd² | 4,046.86 m² |
| 2 acres | 87,120 ft² | 9,680 yd² | 8,093.72 m² |
| 5 acres | 217,800 ft² | 24,200 yd² | 20,234.3 m² |
| 10 acres | 435,600 ft² | 48,400 yd² | 40,468.6 m² |
Frequently Asked Questions
Common questions about acres, acreage calculations, land measurement, and unit conversions.
To calculate the area of a 4-sided irregular lot (quadrilateral), use Bretschneider's formula or the coordinate method. With side lengths only, Brahmagupta's formula provides a close estimate for cyclic quadrilaterals: Area = √((s−a)(s−b)(s−c)(s−d)), where s is the semi-perimeter and a, b, c, d are the four side lengths. For exact results with any quadrilateral, use the coordinate method: plot each corner as (x, y) coordinates and apply the Shoelace formula. The polygon area calculator above supports both methods.
Yes, you can calculate acreage for a 5-sided lot (pentagon) with uneven sides using the coordinate method. Measure each corner of the property as GPS coordinates or plot coordinates, enter them into the polygon area calculator, and the Shoelace formula calculates the exact area in square feet. Divide by 43,560 to convert to acres. For side-length-only input, the calculator approximates the area using the average side length applied to a regular polygon formula.
To find area using coordinates, list the (x, y) position of each vertex (corner) of the polygon in order around the boundary. Then apply the Shoelace formula: Area = ½ × |Σ(xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ)|. This formula works for any polygon shape — regular, irregular, convex, or concave — as long as the vertices are listed in sequential order. The polygon area calculator above automates this calculation and converts the result to acres, hectares, and other units.
The Shoelace formula (also called the Surveyor's formula or Gauss's area formula) calculates the area of any polygon from its vertex coordinates. The formula is: Area = ½ × |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)| for vertices listed in order. It gets its name because the cross-multiplication pattern resembles lacing a shoe. The Shoelace formula is the standard method used in land surveying, GIS software, and CAD applications for computing polygon areas.
Polygon area calculations using the Shoelace formula are mathematically exact when the vertex coordinates are accurate. The accuracy depends entirely on how precisely the corner points are measured. Professional land surveys using total stations achieve accuracy within 0.01 feet (3 mm). GPS measurements from consumer devices are accurate within 3–10 feet (1–3 meters). For legal property boundaries, always use coordinates from a licensed land surveyor's plat map or certified GPS survey.
For a regular hexagon (6 equal sides), use the formula: Area = (3√3 / 2) × s², where s is the side length. For an irregular hexagon with 6 different side lengths, use the coordinate method: measure each corner as an (x, y) point and apply the Shoelace formula. Alternatively, divide the hexagon into triangles, calculate each triangle's area separately, and sum them. The polygon area calculator above handles both regular and irregular hexagons.
A regular polygon has all sides equal in length and all interior angles equal (e.g., a square, equilateral triangle, or regular hexagon). An irregular polygon has sides of different lengths and/or unequal angles (e.g., a rectangular lot that isn't perfectly square, or an L-shaped parcel). Most real-world land parcels are irregular polygons. The polygon area calculator handles both types — use the 'Regular polygon' checkbox for equal sides, or enter each side individually for irregular shapes.
To calculate acreage for an irregular lot: (1) Identify all corners of the property from a survey or plat map, (2) Record each corner as an (x, y) coordinate in feet or meters, (3) Enter the coordinates into a polygon area calculator or apply the Shoelace formula, (4) Divide the result in square feet by 43,560 to get acres. For properties with curved boundaries, approximate the curve with multiple short straight segments. Google Earth Pro can also trace irregular lot boundaries and display the area directly.
You can calculate exact area from side lengths alone only for triangles (3 sides, using Heron's formula) and regular polygons (all sides and angles equal). For irregular polygons with 4 or more sides, side lengths alone do not uniquely determine the shape — the same four sides can form many different quadrilaterals with different areas. For exact results with irregular polygons, you need either coordinates or side lengths plus interior angles. The polygon area calculator provides estimates for side-length-only input and exact results for coordinate input.
A square lot with 4 sides of 200 feet each has an area of 200 × 200 = 40,000 square feet, which equals 0.918 acres (40,000 ÷ 43,560). However, if the 4 sides of 200 feet form a non-square shape (e.g., a rhombus or irregular quadrilateral), the area will be less than 40,000 sq ft. The maximum area for a quadrilateral with a fixed perimeter is always a square. Use the polygon area calculator with coordinates for exact measurements of non-square lots.